From 29eb708670963c0ca5bd315c83a3cec8dafef1a7 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Thu, 20 Feb 2020 13:15:34 +0100
Subject: Commit inicial, primer cuatrimestre.

---
 cyn/n.lyx  |  237 ++++++
 cyn/n1.lyx |  914 +++++++++++++++++++++
 cyn/n2.lyx | 1510 ++++++++++++++++++++++++++++++++++
 cyn/n3.lyx |  725 ++++++++++++++++
 cyn/n4.lyx |  374 +++++++++
 cyn/n5.lyx | 2359 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 cyn/n7.lyx | 2681 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 cyn/n8.lyx |  954 +++++++++++++++++++++
 8 files changed, 9754 insertions(+)
 create mode 100644 cyn/n.lyx
 create mode 100644 cyn/n1.lyx
 create mode 100644 cyn/n2.lyx
 create mode 100644 cyn/n3.lyx
 create mode 100644 cyn/n4.lyx
 create mode 100644 cyn/n5.lyx
 create mode 100644 cyn/n7.lyx
 create mode 100644 cyn/n8.lyx

(limited to 'cyn')

diff --git a/cyn/n.lyx b/cyn/n.lyx
new file mode 100644
index 0000000..56a3b06
--- /dev/null
+++ b/cyn/n.lyx
@@ -0,0 +1,237 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize a5paper
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 0.2cm
+\topmargin 0.7cm
+\rightmargin 0.2cm
+\bottommargin 0.7cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle empty
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Conjuntos y números
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2017}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Curso de conjuntos y números: Apuntes, Juan Jacobo Simón Pinero (Curso 2017–2018
+).
+\end_layout
+
+\begin_layout Chapter
+Conjuntos y elementos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Aplicaciones
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Órdenes en conjuntos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Relaciones de equivalencia
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Conjuntos numéricos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n5.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+El anillo de los números enteros
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n7.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n8.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n1.lyx b/cyn/n1.lyx
new file mode 100644
index 0000000..21cc0c8
--- /dev/null
+++ b/cyn/n1.lyx
@@ -0,0 +1,914 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Podemos definir conjuntos:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Por extensión:
+\series default
+ 
+\begin_inset Formula $A=\{X_{1},\dots,X_{n},\dots\}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Por comprehensión:
+\series default
+ 
+\begin_inset Formula $A=\{X\in B|p(X)\text{ (es verdadera)}\}$
+\end_inset
+
+.
+ Si es obvio quién es 
+\begin_inset Formula $B$
+\end_inset
+
+, se puede omitir.
+\end_layout
+
+\begin_layout Standard
+Cualquiera de ambas escrituras determina un único conjunto.
+ 
+\series bold
+Paradoja de Russell:
+\series default
+ si 
+\begin_inset Formula $\mathcal{U}$
+\end_inset
+
+ es la colección de todos los conjuntos y 
+\begin_inset Formula $A=\{x\in{\cal U}|x\notin x\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $A\in A$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $A\notin A$
+\end_inset
+
+.
+ Lo que ocurre es que 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ no es un conjunto.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Pertenencia:
+\series default
+ 
+\begin_inset Formula $a\in A$
+\end_inset
+
+.
+ Contrario: 
+\begin_inset Formula $a\notin A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Inclusión:
+\series default
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ está contenido, o es un subconjunto, de 
+\series bold
+
+\begin_inset Formula $B$
+\end_inset
+
+
+\series default
+: 
+\begin_inset Formula $A\subseteq B:\iff(a\in A\implies a\in B)$
+\end_inset
+
+.
+ Es transitiva: 
+\begin_inset Formula $A\subseteq B\land B\subseteq C\implies A\subseteq C$
+\end_inset
+
+.
+ Contrario: 
+\begin_inset Formula $A\nsubseteq B$
+\end_inset
+
+.
+ Subconjunto estricto: 
+\begin_inset Formula $A\subsetneq B\iff A\subseteq B\land A\neq B$
+\end_inset
+
+.
+ 
+\begin_inset Formula $A\subset B$
+\end_inset
+
+ es ambiguo, aunque se suele usar como 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Igualdad:
+\series default
+ 
+\begin_inset Formula $A=B:\iff(a\in A\iff a\in B)\iff A\subseteq B\land B\subseteq A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Múltiplos de un número 
+\begin_inset Formula $n$
+\end_inset
+
+ como 
+\begin_inset Formula $n\mathbb{Z}=\{nt|t\in\mathbb{Z}\}=\{nt\}_{t\in\mathbb{Z}}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $m\in n\mathbb{Z}\implies m\mathbb{Z}\subseteq n\mathbb{Z}$
+\end_inset
+
+.
+ Relación 
+\begin_inset Quotes cld
+\end_inset
+
+
+\begin_inset Formula $m$
+\end_inset
+
+ divide a 
+\begin_inset Formula $n$
+\end_inset
+
+
+\begin_inset Quotes crd
+\end_inset
+
+ o 
+\begin_inset Quotes cld
+\end_inset
+
+
+\begin_inset Formula $n$
+\end_inset
+
+ es múltiplo de 
+\begin_inset Formula $m$
+\end_inset
+
+
+\begin_inset Quotes crd
+\end_inset
+
+: 
+\begin_inset Formula $m|n\iff\exists t\in\mathbb{Z}:n=tm$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A=\{x\in B|p(x)\}$
+\end_inset
+
+ y 
+\begin_inset Formula $A'=\{x\in B|p'(x)\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $(p(x)\implies p'(x))\implies A\subseteq A'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+conjunto vacío
+\series default
+ es aquel que no tiene elementos.
+ Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es vacío, entonces 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+, dado que si 
+\begin_inset Formula $A\nsubseteq B$
+\end_inset
+
+ significaría que 
+\begin_inset Formula $\exists a\in A:a\notin B$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $A$
+\end_inset
+
+ no estaría vacío.
+ De aquí podemos deducir que solo hay un conjunto vacío, y lo llamamos 
+\begin_inset Formula $\emptyset$
+\end_inset
+
+.
+ 
+\begin_inset Formula $A=\emptyset:\iff\forall x,x\notin A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El conjunto 
+\begin_inset Formula ${\cal P}(A)=\{B|B\subseteq A\}$
+\end_inset
+
+ es el conjunto de las 
+\series bold
+partes de 
+\begin_inset Formula $A$
+\end_inset
+
+
+\series default
+ o el conjunto 
+\series bold
+potencia
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ También se llama 
+\begin_inset Formula $2^{A}$
+\end_inset
+
+ porque si 
+\begin_inset Formula $A$
+\end_inset
+
+ tiene 
+\begin_inset Formula $n$
+\end_inset
+
+ elementos, 
+\begin_inset Formula ${\cal P}(A)$
+\end_inset
+
+ tiene 
+\begin_inset Formula $2^{n}$
+\end_inset
+
+, de lo que deducimos que 
+\begin_inset Formula $A\neq{\cal P}(A)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Operaciones con subconjuntos
+\end_layout
+
+\begin_layout Standard
+Los 
+\series bold
+diagramas de Venn
+\series default
+ aportan una mejor comprensión de los conjuntos y sus operaciones.
+ Los conjuntos se representan como formas (normalmente círculos y cuadrados),
+ que pueden ir acompañados del nombre del conjunto, y se colorea la parte
+ deseada.
+ Operaciones:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Unión:
+\series default
+ 
+\begin_inset Formula $A\cup B=\{x|x\in A\lor x\in B\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Intersección:
+\series default
+ 
+\begin_inset Formula $A\cap B=\{x|x\in A\land x\in B\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ son 
+\series bold
+disjuntos
+\series default
+ si 
+\begin_inset Formula $A\cap B=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+
+\series bold
+Diferencia de conjuntos:
+\series default
+ 
+\begin_inset Formula $A\backslash B=\{x|x\in A\land x\notin B\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Complemento:
+\series default
+ Si 
+\begin_inset Formula $A\subseteq U$
+\end_inset
+
+, siendo 
+\begin_inset Formula $U$
+\end_inset
+
+ un 
+\begin_inset Quotes cld
+\end_inset
+
+universo
+\begin_inset Quotes crd
+\end_inset
+
+ en el contexto en el que operamos, el complemento de 
+\begin_inset Formula $A$
+\end_inset
+
+ en 
+\begin_inset Formula $U$
+\end_inset
+
+ se define como 
+\begin_inset Formula $A^{\complement}=\overline{A}=U\backslash A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\cap B\subseteq A\subseteq A\cup B$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\forall A\subseteq B,A\cup X\subseteq B\cup X\land A\cap X\subseteq B\cap X$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\subseteq C\land B\subseteq C\implies(A\cup B)\subseteq C$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $(A\cap B)\cap C=A\cap(B\cap C)$
+\end_inset
+
+; 
+\begin_inset Formula $(A\cup B)\cup C=A\cup(B\cup C)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\subseteq B\iff A\cup B=B\iff A\cap B=A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\cup\emptyset=A$
+\end_inset
+
+; 
+\begin_inset Formula $A\cap\emptyset=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $X\subseteq(A\cap B)\iff(X\subseteq A)\land(X\subseteq B)$
+\end_inset
+
+; 
+\begin_inset Formula $(A\cup B)\subseteq X\iff(A\subseteq X)\land(B\subseteq X)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$
+\end_inset
+
+; 
+\begin_inset Formula $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $(A\backslash B)\cup(B\backslash A)=(A\cup B)\backslash(A\cap B)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Leyes de Morgan:
+\series default
+ 
+\begin_inset Formula $(A\cap B)^{\complement}=A^{\complement}\cup B^{\complement}$
+\end_inset
+
+; 
+\begin_inset Formula $(A\cup B)^{\complement}=A^{\complement}\cap B^{\complement}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Familias de conjuntos
+\end_layout
+
+\begin_layout Standard
+Una familia de conjuntos es una colección 
+\begin_inset Formula $\{A_{i}|i\in I\}$
+\end_inset
+
+ donde 
+\begin_inset Formula $I$
+\end_inset
+
+ y 
+\begin_inset Formula $A_{i}$
+\end_inset
+
+ son conjuntos.
+ Si todos los elementos son diferentes, tenemos un conjunto.
+ Algunas definiciones:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Unión arbitraria:
+\series default
+ 
+\begin_inset Formula $\cup{\cal C}=\{x|\exists A\in{\cal C}:x\in A\}$
+\end_inset
+
+; 
+\begin_inset Formula $\cup_{i\in I}A_{i}=\{x|\exists i\in I:x\in A_{i}\}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Intersección arbitraria:
+\series default
+ 
+\begin_inset Formula $\cap{\cal C}=\{x|\forall A\in{\cal C}:x\in A\}$
+\end_inset
+
+; 
+\begin_inset Formula $\cap_{i\in I}A_{i}=\{x|\forall i\in I:x\in A_{i}\}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\cap{\cal C}\subseteq A\subseteq\cup{\cal C}\forall A\in{\cal C}$
+\end_inset
+
+; 
+\begin_inset Formula $\cap A_{i}\subseteq A_{j}\subseteq\cup A_{i}\forall j\in I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\subseteq\cap{\cal C}\iff X\subseteq A\forall A\in{\cal C}$
+\end_inset
+
+; 
+\begin_inset Formula $\cup{\cal C}\subseteq X\iff A\subseteq X\forall A\in{\cal C}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\cup_{i\in I}(A\cap B_{i})=A\cap(\cup_{i\in I}B_{i})$
+\end_inset
+
+; 
+\begin_inset Formula $\cap_{i\in I}(A\cup B_{i})=A\cup(\cap_{i\in I}B_{i})$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $x\in\cup_{i\in I}(A\cap B_{i})\implies\exists i\in I:x\in(A\cap B_{i})\implies(x\in A)\land(x\in B_{i}\subseteq\cup B_{i})$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $x\in A\cap(\cup_{i\in I}B_{i})\implies\exists i:(x\in A\land x\in B_{i})\implies x\in\cup(A\cap B_{i})$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $(\cap A_{i})^{\complement}=\cup A_{i}^{\complement}$
+\end_inset
+
+; 
+\begin_inset Formula $(\cup A_{i})^{\complement}=\cap A_{i}^{\complement}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Pares ordenados, producto cartesiano y relaciones binarias
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+par ordenado
+\series default
+ o 
+\series bold
+pareja ordenada
+\series default
+ formada por 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+ es 
+\begin_inset Formula $(a,b)=\{\{a\},\{a,b\}\}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $(a,b)=(c,d)\iff a=c\land b=d$
+\end_inset
+
+.
+ El 
+\series bold
+producto cartesiano
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ es 
+\begin_inset Formula $A\times B=\{(a,b)|a\in A\land b\in B\}$
+\end_inset
+
+.
+ Este no es asociativo, pues en general, 
+\begin_inset Formula $(A\times B)\times C\neq A\times(B\times C)$
+\end_inset
+
+, pero son biyectivos.
+ Por ahora no tenemos descripción en términos de conjuntos para la expresión
+ 
+\begin_inset Formula $(a,b,c)$
+\end_inset
+
+.
+ Propiedades del producto cartesiano:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\times\emptyset=\emptyset\times A=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\times(B\cup C)=(A\times B)\cup(A\times C)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\times(B\cap C)=(A\times B)\cap(A\times C)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+relación binaria
+\series default
+ o 
+\series bold
+correspondencia
+\series default
+ entre elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ es un subconjunto 
+\begin_inset Formula $R\subseteq A\times B$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $(a,b)\in R$
+\end_inset
+
+, decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ está relacionado con 
+\begin_inset Formula $b$
+\end_inset
+
+, escrito 
+\begin_inset Formula $aRb$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A=B$
+\end_inset
+
+, tenemos una relación en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Definiciones:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Conjunto inicial:
+\series default
+ 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Conjunto final:
+\series default
+ 
+\begin_inset Formula $B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Dominio:
+\series default
+ 
+\begin_inset Formula $\text{Dom}R=\{a\in A|\exists b\in B:(a,b)\in R\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Imagen:
+\series default
+ 
+\begin_inset Formula $\text{Im}R=\{b\in B|\exists a\in A:(a,b)\in R\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Podemos representar las relaciones en gráficas planas.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n2.lyx b/cyn/n2.lyx
new file mode 100644
index 0000000..386c747
--- /dev/null
+++ b/cyn/n2.lyx
@@ -0,0 +1,1510 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una 
+\series bold
+aplicación
+\series default
+ entre dos conjuntos 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ es una relación 
+\begin_inset Formula $f\subseteq A\times B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall a\in A,\exists!b\in B:(a,b)\in f$
+\end_inset
+
+.
+ Escribimos 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ o 
+\begin_inset Formula $A\overset{f}{\longrightarrow}B$
+\end_inset
+
+, y llamamos 
+\begin_inset Formula $b=f(a)\iff(a,b)\in f$
+\end_inset
+
+.
+ Por ejemplo, podemos definir 
+\begin_inset Formula $f:\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(n)=n^{2}$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $f=\{(n,n^{2}):n\in\mathbb{N}\}$
+\end_inset
+
+.
+ Si partimos de una igualdad y queremos interpretarla como la regla de una
+ aplicación, la llamamos 
+\series bold
+función
+\series default
+.
+ Podemos representar una aplicación:
+\end_layout
+
+\begin_layout Enumerate
+Como dos conjuntos representados de forma similar a un diagrama de Euler-Venn,
+ en el que de cada elemento de 
+\begin_inset Formula $A$
+\end_inset
+
+ parte una flecha hacia uno de 
+\begin_inset Formula $B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Como una gráfica, en la que los elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ se representan en el eje horizontal y los de 
+\begin_inset Formula $B$
+\end_inset
+
+ en el eje vertical, y las relaciones se representan con puntos.
+\end_layout
+
+\begin_layout Standard
+Definimos:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Dominio
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+: 
+\begin_inset Formula $\text{Dom}f=A$
+\end_inset
+
+, por lo que el término 
+\begin_inset Quotes cld
+\end_inset
+
+conjunto inicial
+\begin_inset Quotes crd
+\end_inset
+
+ no se usa.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Codominio
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+: 
+\begin_inset Quotes cld
+\end_inset
+
+Conjunto final
+\begin_inset Quotes crd
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Imagen
+\series default
+ o 
+\series bold
+imagen directa
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+: 
+\begin_inset Formula $\text{Im}f=f(A)=\{b\in B:\exists a:f(a)=b\}\subseteq B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Regla de correspondencia
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+: Igualdad 
+\begin_inset Formula $b=f(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $b=f(a)$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\emph on
+una
+\emph default
+ 
+\series bold
+preimagen
+\series default
+ de 
+\begin_inset Formula $b$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ es la 
+\series bold
+imagen
+\series default
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+ley de composición externa
+\series default
+ es una aplicación 
+\begin_inset Formula $B\times A\overset{\circ}{\longrightarrow}A$
+\end_inset
+
+.
+ Una 
+\series bold
+operación binaria
+\series default
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ es una aplicación 
+\begin_inset Formula $A\times A\overset{\circ}{\longrightarrow}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Aplicaciones inyectivas, suprayectivas y biyectivas
+\end_layout
+
+\begin_layout Standard
+La aplicación 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ es:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Inyectiva
+\series default
+ o 
+\series bold
+uno a uno
+\series default
+ si 
+\begin_inset Formula $f(a)=f(b)\implies a=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Suprayectiva
+\series default
+, 
+\series bold
+sobreyectiva
+\series default
+ o 
+\series bold
+exhaustiva
+\series default
+ si 
+\begin_inset Formula $\forall b\in B,\exists a\in A:f(a)=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Biyectiva
+\series default
+ si es inyectiva y suprayectiva.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+restricción de 
+\begin_inset Formula $f$
+\end_inset
+
+ a su imagen
+\series default
+ es una aplicación 
+\begin_inset Formula $\hat{f}:A\rightarrow\text{Im}f$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{f}(a)=f(a)$
+\end_inset
+
+.
+ Se dice que 
+\begin_inset Formula $\hat{f}$
+\end_inset
+
+ 
+\begin_inset Quotes cld
+\end_inset
+
+actúa igual
+\begin_inset Quotes crd
+\end_inset
+
+ que 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Siempre es suprayectiva.
+\end_layout
+
+\begin_layout Section
+Imágenes directas e inversas
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $X\subseteq A$
+\end_inset
+
+, definimos la 
+\series bold
+imagen directa
+\series default
+ de 
+\begin_inset Formula $X$
+\end_inset
+
+ como 
+\begin_inset Formula $f(X)=\{f(x)|x\in X\}$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f(\emptyset)=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Se deriva de que 
+\begin_inset Formula $\emptyset\times B=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $X\subseteq Y\implies f(X)\subseteq f(Y)$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+y\in f(X)\implies\exists x\in X,y\in Y:f(x)=y\implies f(x)\in f(Y)\implies y\in f(Y)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X,Y\subseteq A\implies f(X\cup Y)=f(X)\cup f(Y)$
+\end_inset
+
+; 
+\begin_inset Formula $f\left(\bigcup_{\alpha\in I}X_{\alpha}\right)=\bigcup_{\alpha\in I}f(X_{\alpha})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $y\in f(\cup_{\alpha\in I}X_{\alpha})$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\exists x\in\cup_{\alpha\in I}X_{\alpha}:f(x)=y$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $x\in\cup_{\alpha\in I}X_{\alpha}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\exists\alpha\in I:x\in X_{\alpha}$
+\end_inset
+
+, luego 
+\begin_inset Formula $y\in f(X_{\alpha})\subseteq\cup_{\alpha\in I}f(X_{\alpha})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Considérese 
+\begin_inset Formula $y\in\cup_{\alpha\in I}f(X_{\alpha})$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\exists\alpha\in I:y\in f(X_{\alpha})$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\exists x\in X_{\alpha}:f(x)=y$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x\in\cup_{\alpha\in I}X_{\alpha}$
+\end_inset
+
+, así que 
+\begin_inset Formula $y=f(x)\in f(\cup_{\alpha\in I}X_{\alpha})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $X,Y\subseteq A\implies f(X\cap Y)\subseteq f(X)\cap f(Y)$
+\end_inset
+
+; 
+\begin_inset Formula $f\left(\bigcap_{\alpha\in I}X_{\alpha}\right)\subseteq\bigcap_{\alpha\in I}f(X_{\alpha})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $y\in f(\cap_{\alpha\in I}X_{\alpha})$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\exists x\in\cap_{\alpha\in I}X_{\alpha}:f(x)=y$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $x\in\cap_{\alpha\in I}X_{\alpha}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\forall\alpha\in I,x\in X_{\alpha}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\forall\alpha\in I,\exists x\in X_{\alpha}:f(x)=y$
+\end_inset
+
+.
+ De aquí deducimos que 
+\begin_inset Formula $\forall\alpha\in I,y\in f(X_{\alpha})$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $y\in\cap_{\alpha\in I}f(X_{\alpha})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $Y\subseteq B$
+\end_inset
+
+, definimos la 
+\series bold
+imagen inversa
+\series default
+ de 
+\begin_inset Formula $Y$
+\end_inset
+
+ como 
+\begin_inset Formula $f(Y)^{-1}:=f^{-1}(Y):=\{a\in A|f(a)\in Y\}$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f(\emptyset)^{-1}=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Se deriva de que 
+\begin_inset Formula $A\times\emptyset=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $f(B)^{-1}=A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\subseteq B\implies\left(f(X)^{-1}\right)^{\complement}=f\left(X^{\complement}\right)^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\subseteq Y\subseteq B\implies f(X)^{-1}\subseteq f(Y)^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X,Y\subseteq B\implies f(X\cup Y)^{-1}=f(X)^{-1}\cup f(Y)^{-1}$
+\end_inset
+
+; 
+\begin_inset Formula $f\left(\bigcup_{\alpha\in I}X_{\alpha}\right)^{-1}=\bigcup_{\alpha\in I}f(X_{\alpha})^{-1}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula $x\in f(\bigcup_{\alpha\in I}X_{\alpha})^{-1}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f(x)\in\bigcup_{\alpha\in I}X_{\alpha}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\exists\alpha\in I:f(x)\in X_{\alpha}$
+\end_inset
+
+, de donde 
+\begin_inset Formula $x\in f(X_{\alpha})^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X,Y\subseteq B\implies f(X\cap Y)^{-1}=f(X)^{-1}\cap f(Y)^{-1}$
+\end_inset
+
+; 
+\begin_inset Formula $f\left(\bigcap_{\alpha\in I}Y_{\alpha}\right)^{-1}=\bigcap_{\alpha\in I}f(Y_{\alpha})^{-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $x\in f(\bigcap_{\alpha\in I}Y_{\alpha})^{-1}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f(x)\in\bigcap_{\alpha\in I}Y_{\alpha}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $f(x)\in Y_{\alpha}$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $x\in f(Y_{\alpha})^{-1}$
+\end_inset
+
+, para todo 
+\begin_inset Formula $\alpha\in I$
+\end_inset
+
+.
+ De aquí se tiene que 
+\begin_inset Formula $x\in\bigcap_{\alpha\in I}f(Y_{\alpha})^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $x\in\bigcap_{\alpha\in I}f(Y_{\alpha})^{-1}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x\in f(Y_{\alpha})^{-1}$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $f(x)\in Y_{\alpha}$
+\end_inset
+
+, para todo 
+\begin_inset Formula $\alpha\in I$
+\end_inset
+
+.
+ Esto significa que 
+\begin_inset Formula $f(x)\in\bigcap_{\alpha\in I}Y_{\alpha}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $x\in f(\bigcap_{\alpha\in I}Y_{\alpha})^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ es una aplicación, para todo 
+\begin_inset Formula $X\subseteq A$
+\end_inset
+
+, 
+\begin_inset Formula $X\subseteq f(f(X))^{-1}$
+\end_inset
+
+, y para todo 
+\begin_inset Formula $Y\subseteq B$
+\end_inset
+
+, 
+\begin_inset Formula $f(f(Y)^{-1})\subseteq Y$
+\end_inset
+
+, y ambos contenidos pueden ser estrictos.
+\end_layout
+
+\begin_layout Section
+Composición
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ y 
+\begin_inset Formula $g:B\rightarrow C$
+\end_inset
+
+, definimos la 
+\series bold
+composición de 
+\begin_inset Formula $f$
+\end_inset
+
+ seguida de 
+\begin_inset Formula $g$
+\end_inset
+
+
+\series default
+ como la aplicación 
+\begin_inset Formula $g\circ f:A\rightarrow C$
+\end_inset
+
+ tal que 
+\begin_inset Formula $(g\circ f)(x)=g(f(x))$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{Dom}(g\circ f)=\text{Dom}f$
+\end_inset
+
+ y el codominio de 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es igual al de 
+\begin_inset Formula $g$
+\end_inset
+
+.
+ Además, si 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+, 
+\begin_inset Formula $g:B\rightarrow C$
+\end_inset
+
+ y 
+\begin_inset Formula $h:C\rightarrow D$
+\end_inset
+
+ son aplicaciones, entonces 
+\begin_inset Formula $h\circ(g\circ f)=(h\circ g)\circ f$
+\end_inset
+
+.
+ La demostración parte de la coincidencia entre dominios y codominios que
+ permite considerar las distintas composiciones:
+\begin_inset Formula 
+\[
+(h\circ(g\circ f))(a)=h((g\circ f)(a))=h(g(f(a)))=(h\circ g)(f(a))=((h\circ g)\circ f)(a)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+La composición de aplicaciones inyectivas es inyectiva.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ aplicaciones inyectivas y 
+\begin_inset Formula $a,a'\in A$
+\end_inset
+
+ tales que 
+\begin_inset Formula $(g\circ f)(a)=(g\circ f)(a')$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g(f(a))=g(f(a'))$
+\end_inset
+
+, y como 
+\begin_inset Formula $g$
+\end_inset
+
+ es inyectiva, 
+\begin_inset Formula $f(a)=f(a')$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $a=a'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+La composición de aplicaciones suprayectivas es suprayectiva.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ suprayectivas y 
+\begin_inset Formula $c\in C$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\exists b\in B:g(b)=c$
+\end_inset
+
+ y a su vez 
+\begin_inset Formula $\exists a\in A:f(a)=b$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(g\circ f)(a)=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+La composición de aplicaciones biyectivas es biyectiva.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es inyectiva, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $a,a'\in A$
+\end_inset
+
+ tales que 
+\begin_inset Formula $f(a)=f(a')$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g(f(a))=g(f(a'))$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(g\circ f)(a)=(g\circ f)(a')$
+\end_inset
+
+, y por ello 
+\begin_inset Formula $a=a'$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es suprayectiva, 
+\begin_inset Formula $g$
+\end_inset
+
+ también lo es.
+\begin_inset Newline newline
+\end_inset
+
+Para cualquier 
+\begin_inset Formula $c\in C$
+\end_inset
+
+, 
+\begin_inset Formula $\exists a\in A:(g\circ f)(a)=g(f(a))=c$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $\exists f(a)=b\in B:g(b)=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ y 
+\begin_inset Formula $X\subseteq A$
+\end_inset
+
+, la 
+\series bold
+restricción
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ a 
+\begin_inset Formula $X$
+\end_inset
+
+ es la aplicación 
+\begin_inset Formula $f|_{X}:X\rightarrow B$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f|_{X}(x)=f(x)$
+\end_inset
+
+.
+ También se puede interpretar como que 
+\begin_inset Formula $f|_{X}=f\circ u$
+\end_inset
+
+ con 
+\begin_inset Formula $u:X\rightarrow A$
+\end_inset
+
+ como la 
+\series bold
+aplicación inclusión
+\series default
+, dada por 
+\begin_inset Formula $u(x)=x$
+\end_inset
+
+.
+ Al restringir una aplicación pueden variar sus propiedades.
+\end_layout
+
+\begin_layout Subsection
+Inversa de una aplicación biyectiva
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+aplicación identidad
+\series default
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ como 
+\begin_inset Formula $1_{A}:A\rightarrow A$
+\end_inset
+
+ con 
+\begin_inset Formula $1_{A}(a)=a$
+\end_inset
+
+.
+ Entonces decimos que 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ es una 
+\series bold
+aplicación invertible
+\series default
+ o que tiene 
+\series bold
+inversa
+\series default
+ si existe 
+\begin_inset Formula $g:B\rightarrow A$
+\end_inset
+
+ tal que 
+\begin_inset Formula $g\circ f=1_{A}$
+\end_inset
+
+ y 
+\begin_inset Formula $f\circ g=1_{B}$
+\end_inset
+
+.
+ Ahora supongamos que 
+\begin_inset Formula $g$
+\end_inset
+
+ y 
+\begin_inset Formula $h$
+\end_inset
+
+ son inversas de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Entonces,
+\begin_inset Formula 
+\[
+g=g\circ1_{B}=g\circ(f\circ h)=(g\circ f)\circ h=1_{A}\circ h=h
+\]
+
+\end_inset
+
+Por tanto la inversa de una aplicación es única, y la llamamos 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+.
+ Además 
+\begin_inset Formula $f$
+\end_inset
+
+ es invertible si y sólo si es biyectiva.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $a,a'\in A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $f(a)=f(a')$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f^{-1}(f(a))=f^{-1}(f(a'))$
+\end_inset
+
+, luego 
+\begin_inset Formula $a=a'$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva.
+ Ahora, 
+\begin_inset Formula $\forall b\in B,\exists a=f^{-1}(b)\in A:f(a)=b$
+\end_inset
+
+, por lo que es suprayectiva.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para cada 
+\begin_inset Formula $b\in B$
+\end_inset
+
+ consideremos la imagen inversa 
+\begin_inset Formula $f(\{b\})^{-1}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $f$
+\end_inset
+
+ es suprayectiva, 
+\begin_inset Formula $f(\{b\})^{-1}\neq\emptyset$
+\end_inset
+
+, y si 
+\begin_inset Formula $a,a'\in f(\{b\})^{-1}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $b=f(a)=f(a')$
+\end_inset
+
+, y como es inyectiva, 
+\begin_inset Formula $a=a'$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f(\{b\})^{-1}$
+\end_inset
+
+ tiene un solo elemento.
+ Ahora definimos 
+\begin_inset Formula $g:B\rightarrow A$
+\end_inset
+
+ tal que 
+\begin_inset Formula $g(b)\in f(b)^{-1}$
+\end_inset
+
+.
+ Es inmediato comprobar que 
+\begin_inset Formula $g=f^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son invertibles, 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ también lo es y su inversa es 
+\begin_inset Formula $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
+\end_inset
+
+.
+ Un ejemplo de aplicaciones invertibles son las 
+\series bold
+permutaciones
+\series default
+.
+ Sea 
+\begin_inset Formula $0\neq n\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $A=\{a_{1},\dots,a_{n}\}$
+\end_inset
+
+.
+ Entonces una permutación de 
+\begin_inset Formula $A$
+\end_inset
+
+ es una biyección 
+\begin_inset Formula $\sigma:A\rightarrow A$
+\end_inset
+
+.
+ Se suelen denotar como
+\begin_inset Formula 
+\[
+\sigma:\left(\begin{array}{ccc}
+a_{1} & \dots & a_{n}\\
+\sigma(a_{1}) & \dots & \sigma(a_{n})
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $S(A)$
+\end_inset
+
+ al conjunto de las permutaciones de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A=\{1,\dots,n\}$
+\end_inset
+
+, se escribe como 
+\begin_inset Formula $S_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Producto directo
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $I$
+\end_inset
+
+ un conjunto y 
+\begin_inset Formula $F=\{A_{i}\}_{i\in I}$
+\end_inset
+
+ una familia de conjuntos, se define el 
+\series bold
+producto directo
+\series default
+ de 
+\begin_inset Formula $F$
+\end_inset
+
+ como el conjunto
+\begin_inset Formula 
+\[
+\prod_{i\in I}A_{i}=\left\{ f:I\rightarrow\cup_{i\in I}:f(i)\in A_{i}\forall i\in I\right\} 
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f\in\prod_{i\in I}A_{i}$
+\end_inset
+
+, escribimos 
+\begin_inset Formula $f=(x_{i})_{i\in I}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es finito y se escribe como una lista, podemos escribir el conjunto como
+ 
+\begin_inset Formula $A_{1}\times\cdots\times A_{n}=\{(x_{1},\dots,x_{n}):x_{i}\in A_{i},i=1,\dots,n\}$
+\end_inset
+
+.
+ Si no se quiere escribir el conjunto de índices, este se presupone.
+\end_layout
+
+\begin_layout Standard
+Debemos tener en cuenta que el producto cartesiano se usa en la definición
+ de relación y aplicación, por lo que el producto directo requiere de la
+ definición del cartesiano y no puede sustituirlo, aunque exista una biyección
+ cuando el número de factores es finito y usemos la misma escritura.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $I$
+\end_inset
+
+ y 
+\begin_inset Formula $J$
+\end_inset
+
+ conjuntos y 
+\begin_inset Formula $F=\{A_{i}\}_{i\in I}$
+\end_inset
+
+ y 
+\begin_inset Formula $G=\{B_{j}\}_{j\in J}$
+\end_inset
+
+ familias de conjuntos.
+ Si existe una biyección 
+\begin_inset Formula $\sigma:I\rightarrow J$
+\end_inset
+
+ y un conjunto de biyecciones 
+\begin_inset Formula $\{f_{i}:A_{i}\rightarrow B_{\sigma(i)}\}_{i\in I}$
+\end_inset
+
+, entonces existe una biyección 
+\begin_inset Formula $f:\prod_{i\in I}A_{i}\rightarrow\prod_{j\in J}B_{j}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(x)_{j}=f_{\sigma^{-1}(j)}\left(x_{\sigma^{-1}(j)}\right)$
+\end_inset
+
+ para 
+\begin_inset Formula $x\in\prod_{i\in I}A_{i}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ para cada 
+\begin_inset Formula $x\in\prod_{i\in I}A_{i}$
+\end_inset
+
+ y cada 
+\begin_inset Formula $j\in J$
+\end_inset
+
+ existe un único 
+\begin_inset Formula $f_{\sigma^{-1}(j)}\left(x_{\sigma^{-1}(j)}\right)$
+\end_inset
+
+, de modo que la relación es de aplicación, y debemos ver que es biyectiva.
+ Sea 
+\begin_inset Formula $g:\prod_{j\in J}B_{j}\rightarrow\prod_{i\in I}A_{i}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $g(y)_{i}=f_{i}^{-1}\left(y_{\sigma(i)}\right)$
+\end_inset
+
+ (
+\begin_inset Formula $f_{i}^{-1}:B_{\sigma(i)}\rightarrow A_{i}$
+\end_inset
+
+).
+ Como también es aplicación, debemos probar que sean inversas.
+ Entonces:
+\begin_inset Formula 
+\[
+g(f(x))_{i}=f_{i}^{-1}(f(x)_{\sigma(i)})=f_{i}^{-1}\left(f_{\sigma^{-1}(\sigma(i))}\left(x_{\sigma^{-1}(\sigma(i))}\right)\right)=f_{i}^{-1}(f_{i}(x_{i}))=x_{i}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+De forma análoga, 
+\begin_inset Formula $f(g(y))=y$
+\end_inset
+
+, y como tiene inversa, la aplicación es biyectiva.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Axioma de Elección:
+\series default
+ Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es un conjunto no vacío y 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+ una familia de conjuntos no vacíos, entonces 
+\begin_inset Formula $\prod_{i\in I}A_{i}$
+\end_inset
+
+ es no vacío.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n3.lyx b/cyn/n3.lyx
new file mode 100644
index 0000000..de18e21
--- /dev/null
+++ b/cyn/n3.lyx
@@ -0,0 +1,725 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Relaciones de orden
+\end_layout
+
+\begin_layout Standard
+Una relación 
+\begin_inset Formula $R$
+\end_inset
+
+ en un conjunto 
+\begin_inset Formula $A$
+\end_inset
+
+ se dice que es:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Reflexiva
+\series default
+ si 
+\begin_inset Formula $(a,a)\in R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Transitiva
+\series default
+ si 
+\begin_inset Formula $(a,b),(b,c)\in R\implies(a,c)\in R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Simétrica
+\series default
+ si 
+\begin_inset Formula $(a,b)\in R\implies(b,a)\in R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Antisimétrica
+\series default
+ si 
+\begin_inset Formula $(a,b),(b,a)\in R\implies a=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una relación 
+\begin_inset Formula $\leq$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+de orden parcial
+\series default
+ (o un orden parcial) si es reflexiva, transitiva y antisimétrica.
+ Un ejemplo es el 
+\series bold
+orden lexicográfico
+\series default
+ en 
+\begin_inset Formula $K^{n}$
+\end_inset
+
+: 
+\begin_inset Formula $(x_{1},\dots,x_{n})\leq(y_{1},\dots,y_{n})$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $x_{1}<y_{1}$
+\end_inset
+
+, o 
+\begin_inset Formula $x_{1}=y_{1}$
+\end_inset
+
+ y bien los vectores son de un elemento o bien 
+\begin_inset Formula $(x_{2},\dots,x_{n})\leq(y_{2},\dots,y_{n})$
+\end_inset
+
+ en 
+\begin_inset Formula $K^{n-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+conjunto parcialmente ordenado
+\series default
+ (
+\series bold
+CPO
+\series default
+ o 
+\series bold
+COPO
+\series default
+) es un par 
+\begin_inset Formula $(A,\leq)$
+\end_inset
+
+ donde 
+\begin_inset Formula $A$
+\end_inset
+
+ es un conjunto y 
+\begin_inset Quotes cld
+\end_inset
+
+
+\begin_inset Formula $\leq$
+\end_inset
+
+
+\begin_inset Quotes crd
+\end_inset
+
+ una relación de orden en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Si el contexto no deja dudas, diremos que 
+\begin_inset Formula $A$
+\end_inset
+
+ es un COPO.
+ 
+\series bold
+Notación:
+\series default
+ 
+\begin_inset Formula $a<b:\iff a\leq b\land a\neq b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ es un 
+\series bold
+conjunto totalmente
+\series default
+ o 
+\series bold
+linealmente ordenado
+\series default
+, y 
+\begin_inset Formula $\leq$
+\end_inset
+
+ un 
+\series bold
+orden total
+\series default
+ o 
+\series bold
+lineal
+\series default
+, si se satisface la 
+\series bold
+ley de la tricotomía
+\series default
+, es decir, si dados 
+\begin_inset Formula $a,b\in A$
+\end_inset
+
+, ocurre que 
+\begin_inset Formula $a=b$
+\end_inset
+
+, 
+\begin_inset Formula $a<b$
+\end_inset
+
+ o 
+\begin_inset Formula $b<a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Podemos representar conjuntos ordenados mediante 
+\series bold
+diagramas de Hasse
+\series default
+, también llamados 
+\begin_inset Quotes cld
+\end_inset
+
+
+\lang english
+upward drawing
+\lang spanish
+
+\begin_inset Quotes crd
+\end_inset
+
+ o diagramas de grafo de un orden parcial.
+ Se representan los elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ y se unen con una línea las que tienen relación de equivalencia entre sí,
+ sin contar las que se puedan deducir de la reflexividad o transitividad,
+ y con el elemento mayor situado más arriba.
+ También se pueden representar mediante 
+\series bold
+
+\begin_inset Formula $\zeta$
+\end_inset
+
+-matrices
+\series default
+, matrices 
+\begin_inset Formula $\zeta_{A}$
+\end_inset
+
+ con índices en 
+\begin_inset Formula $A$
+\end_inset
+
+, de forma que
+\begin_inset Formula 
+\[
+\zeta_{a,b}=\begin{cases}
+1 & \text{si }a<b\\
+0 & \text{en otro caso}
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Elementos notables en un COPO
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(A,\leq)$
+\end_inset
+
+ un conjunto parcialmente ordenado y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+máximo
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ cuando 
+\begin_inset Formula $a\geq b\forall b\in A$
+\end_inset
+
+.
+ Si existe, es único, pues si fueran 
+\begin_inset Formula $a,a'\in A$
+\end_inset
+
+ máximos de 
+\begin_inset Formula $A$
+\end_inset
+
+, se tendría que 
+\begin_inset Formula $a\leq a'$
+\end_inset
+
+ y 
+\begin_inset Formula $a'\leq a$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $a=a'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+mínimo
+\series default
+ o 
+\series bold
+primer elemento
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ cuando 
+\begin_inset Formula $a\leq b\forall b\in A$
+\end_inset
+
+.
+ Si existe, es único, y la demostración es análoga.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es un 
+\series bold
+elemento maximal
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ cuando 
+\begin_inset Formula $b\geq a\implies b=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es un 
+\series bold
+elemento minimal
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ cuando 
+\begin_inset Formula $b\leq a\implies b=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Además, si 
+\begin_inset Formula $B\subseteq A$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+cota superior
+\series default
+ de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $a\geq b\forall b\in B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+cota inferior
+\series default
+ de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $a\leq b\forall b\in B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+supremo
+\series default
+ o 
+\series bold
+extremo superior
+\series default
+ de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ si es el mínimo de las cotas superiores de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Si existe es único, pues el mínimo de las cuotas superiores, al ser un
+ mínimo, es único.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+ínfimo
+\series default
+ o 
+\series bold
+extremo inferior
+\series default
+ de 
+\series bold
+
+\begin_inset Formula $B$
+\end_inset
+
+
+\series default
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ si es el máximo de las cotas inferiores de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Si existe es único, por razonamiento análogo al anterior.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, 
+\begin_inset Formula $b$
+\end_inset
+
+ es máximo de 
+\begin_inset Formula $B$
+\end_inset
+
+ si y sólo si es el supremo de 
+\begin_inset Formula $B$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Al ser máximo se tiene que 
+\begin_inset Formula $b'\geq b\forall b\in B$
+\end_inset
+
+ y por tanto también es cota superior, pero si hubiera una cota superior
+ menor, a la que llamaremos 
+\begin_inset Formula $c$
+\end_inset
+
+, entonces 
+\begin_inset Formula $c<b\in B$
+\end_inset
+
+ y por tanto no es cota superior
+\begin_inset Formula $\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Al ser supremo, es cota superior, por lo que 
+\begin_inset Formula $a\geq b\forall b\in B$
+\end_inset
+
+.
+ Si a esto le unimos que 
+\begin_inset Formula $a\in B$
+\end_inset
+
+, tenemos la definición de máximo.
+\end_layout
+
+\begin_layout Standard
+Esta propiedad se cumple de forma análoga si en vez del máximo y el supremo
+ tomamos el mínimo y el ínfimo.
+\end_layout
+
+\begin_layout Section
+Conjuntos bien ordenados
+\end_layout
+
+\begin_layout Standard
+Un CPO es 
+\series bold
+bien ordenado
+\series default
+ si todo subconjunto suyo no vacío tiene primer elemento.
+ Todo conjunto bien ordenado es totalmente ordenado.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $A$
+\end_inset
+
+ bien ordenado y 
+\begin_inset Formula $B=\{a,b\}\subseteq A$
+\end_inset
+
+, como 
+\begin_inset Formula $\{a,b\}\neq\emptyset$
+\end_inset
+
+, tiene primer elemento, de lo que se desprende la tricotomía.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Principio de la Buena Ordenación:
+\series default
+ Si 
+\begin_inset Formula $A\neq\emptyset$
+\end_inset
+
+, existe un orden 
+\begin_inset Formula $\leq$
+\end_inset
+
+ tal que 
+\begin_inset Formula $(A,\leq)$
+\end_inset
+
+ es un conjunto bien ordenado.
+ Esto es equivalente al Axioma de Elección.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n4.lyx b/cyn/n4.lyx
new file mode 100644
index 0000000..50a4550
--- /dev/null
+++ b/cyn/n4.lyx
@@ -0,0 +1,374 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
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+\graphics default
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+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una relación es 
+\series bold
+de equivalencia
+\series default
+ si es reflexiva, simétrica y transitiva.
+ Si 
+\begin_inset Formula $(a,b)\in R$
+\end_inset
+
+, escribimos 
+\begin_inset Formula $aRb$
+\end_inset
+
+, 
+\begin_inset Formula $a\sim_{R}b$
+\end_inset
+
+ o, si no causa confusión 
+\begin_inset Formula $a\sim b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Clases de equivalencia
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $A\neq\emptyset$
+\end_inset
+
+ y 
+\begin_inset Formula $R$
+\end_inset
+
+ una relación de equivalencia en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Para cada 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, su clase de equivalencia es 
+\begin_inset Formula $[a]=\{b\in A:a\sim b\}$
+\end_inset
+
+.
+ Entonces:
+\begin_inset Formula 
+\[
+[a]\cap[b]\neq\emptyset\iff a\sim_{R}b\iff[a]=[b]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ 
+\begin_inset Formula $x\in[a]\cap[b]\implies a\sim x\land x\sim b\implies a\sim b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Por hipótesis 
+\begin_inset Formula $a\sim b$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x\in[a]\implies x\sim a\implies x\sim b\implies x\in[b]$
+\end_inset
+
+.
+ Análogamente, 
+\begin_inset Formula $y\in[b]\implies y\in[a]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula $(a,a)\in R\implies a\in[a]=[b]\implies[a]\cap[b]\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $C$
+\end_inset
+
+ es una clase de equivalencia y 
+\begin_inset Formula $a\in C$
+\end_inset
+
+ entonces 
+\begin_inset Formula $[a]=C$
+\end_inset
+
+, y decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ es un 
+\series bold
+representante
+\series default
+ de 
+\begin_inset Formula $C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+El conjunto cociente y la proyección canónica
+\end_layout
+
+\begin_layout Standard
+Se define el 
+\series bold
+conjunto cociente
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ respecto de la relación 
+\begin_inset Formula $R$
+\end_inset
+
+ como el conjunto de las clases de equivalencia de los elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ respecto de 
+\begin_inset Formula $R$
+\end_inset
+
+, y se denota 
+\begin_inset Formula $A/R$
+\end_inset
+
+, 
+\begin_inset Formula $A/\sim_{R}$
+\end_inset
+
+, 
+\begin_inset Formula $A/\sim$
+\end_inset
+
+ o 
+\begin_inset Formula $\frac{A}{\sim}$
+\end_inset
+
+.
+ Calcular los conjuntos cociente consiste en dar un 
+\series bold
+juego completo de representantes
+\series default
+, es decir, describir un conjunto 
+\begin_inset Formula $R$
+\end_inset
+
+ con uno y solo un representante de cada clase de equivalencia (
+\series bold
+conjunto irredundante de representantes
+\series default
+ de las clases de equivalencia).
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+proyección canónica
+\series default
+ a la aplicación 
+\begin_inset Formula $\eta_{R}:A\rightarrow A/R$
+\end_inset
+
+ con 
+\begin_inset Formula $a\mapsto[a]$
+\end_inset
+
+.
+ Siempre es suprayectiva, por la definición de 
+\begin_inset Formula $A/R$
+\end_inset
+
+, y solo es inyectiva cuando 
+\begin_inset Formula $R$
+\end_inset
+
+ es la igualdad.
+\end_layout
+
+\begin_layout Section
+Relaciones de equivalencia y particiones
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ e 
+\begin_inset Formula $I$
+\end_inset
+
+ conjuntos y 
+\begin_inset Formula $P=\{B_{i}\}_{i\in I}$
+\end_inset
+
+ una familia de subconjuntos de 
+\begin_inset Formula $A$
+\end_inset
+
+, decimos que 
+\begin_inset Formula $P$
+\end_inset
+
+ forma una 
+\series bold
+partición
+\series default
+ para 
+\begin_inset Formula $A$
+\end_inset
+
+ si se verifica que 
+\begin_inset Formula $B_{i}\cap B_{j}=\emptyset\iff i\neq j$
+\end_inset
+
+ y 
+\begin_inset Formula $\bigcup_{i\in I}B_{i}=A$
+\end_inset
+
+.
+ Toda relación de equivalencia induce una partición, pues 
+\begin_inset Formula $[a]\cap[b]=\emptyset\iff a\not\sim b$
+\end_inset
+
+, lo que se obtiene de las propiedades de las clases de equivalencia, y
+ 
+\begin_inset Formula $\cup_{[a]\in A/\sim}[a]=A$
+\end_inset
+
+, pues 
+\begin_inset Formula $b\sim b\implies b\in[b]\subseteq\cup_{[a]\in A/\sim}[a]$
+\end_inset
+
+.
+ Del mismo modo, toda partición 
+\begin_inset Formula $\{C_{i}\}_{i\in I}$
+\end_inset
+
+ en 
+\begin_inset Formula $A$
+\end_inset
+
+ determina una clase de equivalencia, definida por 
+\begin_inset Formula $a\sim b:\iff\exists i\in I:a,b\in C_{i}$
+\end_inset
+
+.
+ Solo quedaría probar que esta es una relación de equivalencia y las clases
+ de equivalencia son las 
+\begin_inset Formula $C_{i}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n5.lyx b/cyn/n5.lyx
new file mode 100644
index 0000000..0315b5a
--- /dev/null
+++ b/cyn/n5.lyx
@@ -0,0 +1,2359 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Dos conjuntos son 
+\series bold
+equipotentes
+\series default
+ si existe una aplicación biyectiva entre ellos.
+ Al ser una relación reflexiva, simétrica y transitiva, podemos agrupar
+ a los conjuntos en 
+\begin_inset Quotes cld
+\end_inset
+
+clases de equipotencia
+\begin_inset Quotes crd
+\end_inset
+
+ que llamamos 
+\series bold
+cardinales
+\series default
+, y representamos con 
+\begin_inset Formula $|A|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un conjunto 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+infinito
+\series default
+ si existe 
+\begin_inset Formula $B\subsetneq A$
+\end_inset
+
+ equipotente a 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ En caso contrario es 
+\series bold
+finito
+\series default
+.
+ Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es finito, 
+\begin_inset Formula $f:A\rightarrow A$
+\end_inset
+
+ es inyectiva si y sólo si es suprayectiva.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Se 
+\begin_inset Formula $B=\text{Im}f\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\hat{f}:A\rightarrow B$
+\end_inset
+
+ la restricción a la imagen de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Esta es entonces biyectiva, y como 
+\begin_inset Formula $A$
+\end_inset
+
+ es finito, el subconjunto 
+\begin_inset Formula $B$
+\end_inset
+
+ no puede ser propio, por lo que es 
+\begin_inset Formula $B=A$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\text{Im}f=A$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es suprayectiva.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para cualquier 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ se tiene que 
+\begin_inset Formula $f(\{a\})^{-1}\neq\emptyset$
+\end_inset
+
+, por lo que existe 
+\begin_inset Formula $g:A\rightarrow A\in\prod_{a\in A}f(\{a\})^{-1}$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $f(g(a))=a\forall a\in A$
+\end_inset
+
+, es decir, 
+\begin_inset Formula $f\circ g=1_{A}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $g$
+\end_inset
+
+ es inyectiva y, por la implicación anterior, suprayectiva.
+ Si 
+\begin_inset Formula $a_{1},a_{2}\in A$
+\end_inset
+
+ verifican 
+\begin_inset Formula $f(a_{1})=f(a_{2})$
+\end_inset
+
+ entonces existen, por la suprayectividad de 
+\begin_inset Formula $g$
+\end_inset
+
+, 
+\begin_inset Formula $b_{1},b_{2}\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $g(b_{1})=a_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $g(b_{2})=a_{2}$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $b_{1}=f(g(b_{1}))=f(a_{1})=f(a_{2})=f(g(b_{2}))=b_{2}$
+\end_inset
+
+, de donde 
+\begin_inset Formula $a_{1}=g(b_{1})=g(b_{2})=a_{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva.
+\end_layout
+
+\begin_layout Standard
+Igualmente, si 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ son conjuntos finitos con 
+\begin_inset Formula $|A|=|B|$
+\end_inset
+
+, entonces 
+\begin_inset Formula $g:A\rightarrow B$
+\end_inset
+
+ es inyectiva si y sólo si es suprayectiva.
+ 
+\series bold
+Demostración:
+\series default
+ Al existir una biyección 
+\begin_inset Formula $h:B\rightarrow A$
+\end_inset
+
+, podemos definir 
+\begin_inset Formula $f=h\circ g:A\rightarrow A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $g$
+\end_inset
+
+ es inyectiva, 
+\begin_inset Formula $f$
+\end_inset
+
+ también, por lo que 
+\begin_inset Formula $f$
+\end_inset
+
+ es suprayectiva y 
+\begin_inset Formula $g=h^{-1}\circ f$
+\end_inset
+
+ también.
+ El recíproco se prueba de forma análoga.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+, si 
+\begin_inset Formula $A$
+\end_inset
+
+ es infinito, 
+\begin_inset Formula $B$
+\end_inset
+
+ también lo es.
+ 
+\series bold
+Demostración:
+\series default
+ Existe 
+\begin_inset Formula $A_{0}\subsetneq A$
+\end_inset
+
+ y 
+\begin_inset Formula $f:A_{0}\rightarrow A$
+\end_inset
+
+ biyectiva.
+ Sea entonces 
+\begin_inset Formula $B_{0}=A_{0}\dot{\cup}(B\backslash A)\subsetneq B$
+\end_inset
+
+, basta construir una biyección 
+\begin_inset Formula $f':B_{0}\rightarrow B$
+\end_inset
+
+ con 
+\begin_inset Formula $x\in A_{0}\mapsto f(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in B\backslash A\mapsto x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Números naturales
+\end_layout
+
+\begin_layout Standard
+Un cardinal es finito si tiene un representante finito.
+ De lo contrario es infinito.
+ Llamamos 
+\series bold
+números naturales
+\series default
+ (
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+) a la colección de cardinales finitos.
+ El 
+\series bold
+axioma del infinito
+\series default
+ afirma que esta colección es un conjunto.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $n=|A|$
+\end_inset
+
+, llamamos 
+\series bold
+sucesor
+\series default
+ de 
+\begin_inset Formula $n$
+\end_inset
+
+ a 
+\begin_inset Formula $n^{*}=|A\cup\{x\}|$
+\end_inset
+
+ con 
+\begin_inset Formula $x\notin A$
+\end_inset
+
+.
+ Tenemos que 
+\begin_inset Formula $n^{*}\in\mathbb{N}$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $n^{*}=n+1$
+\end_inset
+
+.
+ Podemos entonces definir 
+\begin_inset Formula $\sigma:\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\sigma(n)=n^{*}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ es inyectiva pero no suprayectiva y por tanto 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ es infinito.
+ Vemos que 
+\begin_inset Formula $0=|\emptyset|$
+\end_inset
+
+ es el único número natural que no es sucesor de ningún otro.
+ Escribimos 
+\begin_inset Formula $\mathbb{N}^{*}=\mathbb{N}\backslash\{0\}$
+\end_inset
+
+, y entonces podemos definir intuitivamente la aplicación 
+\series bold
+antecesor
+\series default
+ como 
+\begin_inset Formula $\hat{\sigma}^{-1}:\mathbb{N}^{*}\rightarrow\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $|A|\leq|B|\iff\exists f:A\rightarrow B\text{ inyectiva}$
+\end_inset
+
+, y vemos que 
+\begin_inset Formula $(\mathbb{N},\leq)$
+\end_inset
+
+ es bien ordenado.
+ Entonces 
+\begin_inset Formula $n^{*}=\min\{x\in\mathbb{N}|n<x\}$
+\end_inset
+
+ y por tanto, si 
+\begin_inset Formula $a,n\in\mathbb{N}$
+\end_inset
+
+ son tales que 
+\begin_inset Formula $n\leq a\leq n^{*}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a=n$
+\end_inset
+
+ o 
+\begin_inset Formula $a=n^{*}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $M_{n}=\{x\in\mathbb{N}|n<x\}$
+\end_inset
+
+ y 
+\begin_inset Formula $a=\min M_{n}$
+\end_inset
+
+.
+ Sabemos que existen 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ representantes respectivos de 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $n$
+\end_inset
+
+ junto con 
+\begin_inset Formula $f:N\rightarrow A$
+\end_inset
+
+ inyectiva pero no suprayectiva.
+ Entonces existe 
+\begin_inset Formula $x\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $x\notin\text{Im}f$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $g:N\cup\{N\}\rightarrow A$
+\end_inset
+
+ con 
+\begin_inset Formula $g(b)=f(b)$
+\end_inset
+
+ para 
+\begin_inset Formula $b\in N$
+\end_inset
+
+ y 
+\begin_inset Formula $g(N)=x$
+\end_inset
+
+, podemos comprobar que 
+\begin_inset Formula $g$
+\end_inset
+
+ es inyectiva y por tanto 
+\begin_inset Formula $n^{*}\leq a$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $n^{*}=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+principio de inducción en los números naturales
+\series default
+ afirma que si 
+\begin_inset Formula $A\subseteq\mathbb{N}$
+\end_inset
+
+ cumple que 
+\begin_inset Formula $0\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in A\implies n^{*}\in A$
+\end_inset
+
+ entonces 
+\begin_inset Formula $A=\mathbb{N}$
+\end_inset
+
+.
+ Esto puede modificarse tomando que para 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, si 
+\begin_inset Formula $k\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $k\leq n\in A\implies n^{*}\in A$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\{n\in\mathbb{N}|n\geq k\}\subseteq A$
+\end_inset
+
+.
+ La 
+\series bold
+inducción matemática
+\series default
+ es un método de demostración consistente en demostrar la validez de la
+ propiedad 
+\begin_inset Formula $P$
+\end_inset
+
+ en 
+\begin_inset Formula $k$
+\end_inset
+
+ y luego probar la validez de 
+\begin_inset Formula $P(n+1)$
+\end_inset
+
+ suponiendo la de 
+\begin_inset Formula $P(n)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+También, dados 
+\begin_inset Formula $A\subseteq\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $k\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall m\in\mathbb{N},(k\leq m<n\implies m\in A)\implies n\in A$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $\{n\in\mathbb{N}|n\geq k\}\subseteq A$
+\end_inset
+
+.
+ La aplicación de esto se conoce como 
+\series bold
+inducción matemática fuerte
+\series default
+.
+ El principio de inducción, el del buen orden y el axioma de elección son
+ equivalentes.
+\end_layout
+
+\begin_layout Standard
+Todo conjunto finito totalmente ordenado está bien ordenado y tiene máximo
+ y mínimo.
+ Por otro lado, 
+\begin_inset Formula $\mathbb{N}_{n}=\{x\in\mathbb{N}|1\leq x\leq n\}$
+\end_inset
+
+ cumple que 
+\begin_inset Formula $|\mathbb{N}_{n}|=|\{1,\dots,n\}|=n$
+\end_inset
+
+ y por tanto es finito.
+\end_layout
+
+\begin_layout Standard
+El conjunto de números naturales que hemos construido satisface los 
+\series bold
+axiomas de Peano
+\series default
+:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $0\in\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\exists\sigma:\mathbb{N}\rightarrow\mathbb{N}\text{ inyectiva}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $0\notin\text{Im}\sigma$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Se cumple el principio de inducción.
+\end_layout
+
+\begin_layout Standard
+Cualquier conjunto que cumpla estas condiciones es esencialmente igual a
+ 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+, lo que se conoce como 
+\series bold
+unicidad del sistema de Peano
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+suma
+\series default
+ como 
+\begin_inset Formula $+:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $n+0=n$
+\end_inset
+
+ y 
+\begin_inset Formula $n+m^{*}=(n+m)^{*}$
+\end_inset
+
+.
+ Esta cumple que 
+\begin_inset Formula $(n+1)+m=n+(m+1)$
+\end_inset
+
+, y verifica las propiedades de 
+\series bold
+conmutatividad
+\series default
+, 
+\series bold
+asociatividad
+\series default
+ y 
+\series bold
+cancelación
+\series default
+ (
+\begin_inset Formula $a+c=b+c\implies a=b$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Definimos el 
+\series bold
+producto
+\series default
+ como 
+\begin_inset Formula $\cdot:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $n\cdot0=0$
+\end_inset
+
+ y 
+\begin_inset Formula $n\cdot m^{*}=n\cdot m+n$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $n\cdot m=nm$
+\end_inset
+
+.
+ Este cumple que 
+\begin_inset Formula $(n+1)m=nm+m$
+\end_inset
+
+, y verifica las propiedades de 
+\series bold
+conmutatividad
+\series default
+, 
+\series bold
+asociatividad
+\series default
+, 
+\series bold
+distributividad
+\series default
+ respecto de la suma y 
+\series bold
+cancelación
+\series default
+ (
+\begin_inset Formula $nm=0\iff n=0\lor m=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema
+\series default
+ para la relación del orden y las operaciones aritméticas:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\leq b\iff\exists u\in\mathbb{N}:a+u=b$
+\end_inset
+
+; 
+\begin_inset Formula $a\leq a+u\forall u\in\mathbb{N}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula $B=\{n\in\mathbb{N}|a+n>b\}$
+\end_inset
+
+ y como 
+\begin_inset Formula $b^{*}\in B$
+\end_inset
+
+ entonces 
+\begin_inset Formula $B\neq\emptyset$
+\end_inset
+
+, por lo que existe 
+\begin_inset Formula $c:=\min B$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $u\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $u^{*}=c$
+\end_inset
+
+.
+ De aquí, 
+\begin_inset Formula $a+u\leq b<a+u^{*}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $a+u=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\leq b\implies a+c\leq b+c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\leq b\implies ac\leq bc$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $a+u=b$
+\end_inset
+
+, llamamos 
+\begin_inset Formula $u=b-a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Números enteros
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+números enteros
+\series default
+ al conjunto cociente 
+\begin_inset Formula $\mathbb{Z}=\mathbb{N}\times\mathbb{N}/\sim$
+\end_inset
+
+ con
+\begin_inset Formula 
+\[
+(a,b)\sim(n,m)\iff a+m=b+n
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Tenemos entonces que 
+\begin_inset Formula $\{(a,0)\}_{a\in\mathbb{N}}\dot{\cup}\{(0,b)\}_{b\in\mathbb{N}^{*}}$
+\end_inset
+
+ es un conjunto irredundante de representantes.
+ Así, si 
+\begin_inset Formula $n\geq m$
+\end_inset
+
+, 
+\begin_inset Formula $(n,m)\in[(n-m,0)]$
+\end_inset
+
+, y si 
+\begin_inset Formula $n<m$
+\end_inset
+
+, 
+\begin_inset Formula $(n,m)\in[(0,m-n)]$
+\end_inset
+
+.
+ Denotamos con 
+\begin_inset Formula $n$
+\end_inset
+
+ a 
+\begin_inset Formula $[(n,0)]$
+\end_inset
+
+ y los identificamos con los naturales, y denotamos con 
+\begin_inset Formula $-n$
+\end_inset
+
+ a 
+\begin_inset Formula $[(0,n)]$
+\end_inset
+
+.
+ Definimos también 
+\begin_inset Formula $\mathbb{Z}^{+}=\{n\in\mathbb{Z}|0\neq n\in\mathbb{N}\}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Z}^{-}=\{-n\in\mathbb{Z}|0\neq n\in\mathbb{N}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{Z}^{*}=\mathbb{Z}\backslash\{0\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $[(a,b)]\leq[(m,n)]\iff a+n\leq b+m$
+\end_inset
+
+, y de aquí que 
+\begin_inset Formula $(\mathbb{Z},\leq)$
+\end_inset
+
+ es un conjunto totalmente ordenado en el que todo entero tiene predecesor
+ y sucesor.
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+suma
+\series default
+ como 
+\begin_inset Formula $+:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $[(a,b)]+[(m,n)]=[(a+m,b+n)]$
+\end_inset
+
+.
+ Esta está bien definida y verifica las propiedades conmutativa, asociativa,
+ existencia de 
+\series bold
+neutro
+\series default
+ 
+\begin_inset Formula $0=[(0,0)]$
+\end_inset
+
+ (
+\begin_inset Formula $\forall a\in\mathbb{Z},a+0=0$
+\end_inset
+
+) y existencia de 
+\series bold
+opuesto
+\series default
+ o 
+\series bold
+inverso bajo la suma
+\series default
+ (
+\begin_inset Formula $\forall a\in\mathbb{Z},\exists a^{\prime}:a+a^{\prime}=0$
+\end_inset
+
+).
+ 
+\series bold
+Demostración
+\series default
+ de que está bien definida.
+ Sean 
+\begin_inset Formula $a,a',b,b',m,m',n,n'\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $[(a,b)]=[(a',b')]$
+\end_inset
+
+ y 
+\begin_inset Formula $[(m,n)]=[(m',n')]$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $a+b'=b+a'$
+\end_inset
+
+ y 
+\begin_inset Formula $m+n'=n+m'$
+\end_inset
+
+, de donde 
+\begin_inset Formula $a+b'+m+n'=b+a'+n+m'$
+\end_inset
+
+, luego 
+\begin_inset Formula $(a+m)+(b'+n')=(a'+m')+(b+n)$
+\end_inset
+
+ y 
+\begin_inset Formula $[(a+m,b+n)]=[(a'+m',b'+n')]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos el 
+\series bold
+producto
+\series default
+ como 
+\begin_inset Formula $\cdot:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $[(a,b)]\cdot[(m,n)]=[(am+bn,an+bm)]$
+\end_inset
+
+.
+ Este está bien definido y verifica las propiedades conmutativa, asociativa,
+ distributiva respecto a la suma y existencia de neutro 
+\begin_inset Formula $1=[(1,0)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Números racionales
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+números racionales
+\series default
+ al conjunto cociente 
+\begin_inset Formula $\mathbb{Q}=\mathbb{Z}\times\mathbb{Z}^{*}/\sim$
+\end_inset
+
+ con
+\begin_inset Formula 
+\[
+[(a,b)]\sim[(n,m)]\iff am=bn
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Identificamos los enteros con los 
+\begin_inset Formula $[(n,1)]$
+\end_inset
+
+, escribimos 
+\begin_inset Formula $\frac{m}{n}:=[(m,n)]$
+\end_inset
+
+ y denotamos con 
+\begin_inset Formula $m$
+\end_inset
+
+ a 
+\begin_inset Formula $\frac{m}{1}=[(m,1)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $\frac{n}{m}\leq\frac{a}{b}\iff nmb^{2}\leq abm^{2}$
+\end_inset
+
+, y decimos que un racional es 
+\series bold
+positivo
+\series default
+ si es mayor que 0 y 
+\series bold
+negativo
+\series default
+ si es menor.
+ Si 
+\begin_inset Formula $m$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ tienen el mismo signo, podemos considerar 
+\begin_inset Formula $\frac{n}{m}\leq\frac{a}{b}\iff nb\leq ma$
+\end_inset
+
+.
+ Se tiene que 
+\begin_inset Formula $(\mathbb{Q},\leq)$
+\end_inset
+
+ es un conjunto totalmente ordenado.
+ 
+\series bold
+Demostración:
+\series default
+ Dados 
+\begin_inset Formula $\frac{n}{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{a}{b}$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $nb=ma$
+\end_inset
+
+, 
+\begin_inset Formula $nb>ma$
+\end_inset
+
+ o 
+\begin_inset Formula $nb<ma$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos la suma como 
+\begin_inset Formula $+:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{a}{b}+\frac{m}{n}=\frac{an+bm}{bn}$
+\end_inset
+
+.
+ Esta está bien definida, y verifica las propiedades de conmutatividad,
+ asociatividad, existencia de neutro 
+\begin_inset Formula $0=[(0,1)]$
+\end_inset
+
+ y existencia de opuesto.
+ Además, 
+\begin_inset Formula $\frac{-n}{m}=\frac{n}{-m}=-\frac{n}{m}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos el producto como 
+\begin_inset Formula $\cdot:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{a}{b}\cdot\frac{m}{n}=\frac{am}{bn}$
+\end_inset
+
+.
+ Este está bien definido y verifica las propiedades de conmutatividad, asociativ
+idad, existencia de neutro 
+\begin_inset Formula $1=[(1,1)]$
+\end_inset
+
+ y existencia de inverso para todo racional no cero (
+\begin_inset Formula $\forall\frac{m}{n}\in\mathbb{Q},\frac{m}{n}\cdot\frac{n}{m}=1$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Una sucesión de números naturales 
+\begin_inset Formula $(a_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ (o de cualquier subconjunto de 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+) es 
+\series bold
+eventualmente periódica
+\series default
+ si 
+\begin_inset Formula $\exists m\in\mathbb{N},q\in\mathbb{N}^{*}:\forall i\geq m,a_{i}=a_{i+q}$
+\end_inset
+
+.
+ Al menor 
+\begin_inset Formula $m$
+\end_inset
+
+ que satisface la condición se le llama 
+\series bold
+término inicial del período
+\series default
+, y al menor 
+\begin_inset Formula $q$
+\end_inset
+
+, 
+\series bold
+período
+\series default
+.
+ Una sucesión eventualmente periódica con 
+\begin_inset Formula $p=1$
+\end_inset
+
+ se dice que es 
+\series bold
+eventualmente constante
+\series default
+\SpecialChar endofsentence
+ Por otro lado, una sucesión de naturales 
+\begin_inset Formula $(a_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ es 
+\series bold
+decimal
+\series default
+ si 
+\begin_inset Formula $a_{n}\in\{0,\dots,9\}$
+\end_inset
+
+ para 
+\begin_inset Formula $n>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema:
+\series default
+ Para todo 
+\begin_inset Formula $\alpha\in\mathbb{Q}$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha\geq0$
+\end_inset
+
+, existe una única sucesión decimal eventualmente periódica de naturales
+ 
+\begin_inset Formula $(a_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $0\leq\alpha-a_{0}-\frac{a_{1}}{10}-\dots-\frac{a_{n}}{10^{n}}<\frac{1}{10^{n}}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+.
+ Esta relación determina una biyección entre los racionales positivos y
+ las sucesiones decimales eventualmente periódicas que no son eventualmente
+ constantes con término inicial 9.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración.
+
+\series default
+ Tomamos 
+\begin_inset Formula $\alpha=\frac{k}{d}$
+\end_inset
+
+ con 
+\begin_inset Formula $k\geq0$
+\end_inset
+
+, 
+\begin_inset Formula $d>0$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{mcd}(k,d)=1$
+\end_inset
+
+ y definimos 
+\begin_inset Formula $a_{0}=E(\alpha)$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{0}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\alpha=a_{0}+\frac{r_{0}}{d}$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $0\leq r_{0}<d$
+\end_inset
+
+.
+ Definimos entonces por recurrencia 
+\begin_inset Formula $a_{n+1}=E(\frac{10r_{n}}{d})$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{10r_{n}}{d}=a_{n+1}+\frac{r_{n+1}}{d}$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $0\leq r_{n+1}<d$
+\end_inset
+
+, y también 
+\begin_inset Formula $S_{n}=a_{0}+a_{1}10^{-1}+\dots+a_{n}10^{-n}$
+\end_inset
+
+.
+ A continuación probamos las siguientes afirmaciones:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Decimal:
+\series default
+ 
+\begin_inset Formula $0\leq a_{n}<10$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+0\leq a_{n+1}\leq a_{n+1}+\frac{r_{n+1}}{d}=\frac{10r_{n}}{d}<10
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Lema:
+\series default
+ 
+\begin_inset Formula $\alpha=S_{n}+\frac{r_{n}}{d}10^{-n}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Para 
+\begin_inset Formula $n=0$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha=a_{0}+\frac{r_{0}}{d}$
+\end_inset
+
+.
+ Ahora asumimos que esto se cumple para un cierto 
+\begin_inset Formula $n$
+\end_inset
+
+ y demostramos que se cumple también para 
+\begin_inset Formula $n+1$
+\end_inset
+
+: 
+\begin_inset Formula 
+\[
+\alpha=S_{n}+\frac{10r_{n}}{d}10^{-(n+1)}=S_{n}+\left(a_{n+1}+\frac{r_{n+1}}{d}\right)10^{-(n+1)}=S_{n+1}+\frac{r_{n+1}}{d}10^{-(n+1)}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Aproximación:
+\series default
+ 
+\begin_inset Formula $0\leq\alpha-S_{n}<10^{n}$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+0\leq\alpha-S_{n+1}=\frac{r_{n+1}}{d}10^{-(n+1)}<10^{-(n+1)}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Unicidad:
+\series default
+ 
+\begin_inset Formula $a_{n+1}=E(10^{n+1}(\alpha-S_{n}))$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+a_{n+1}=E\left(\frac{10r_{n}}{d}\right)=E\left(10^{n+1}10^{-n}\frac{r_{n}}{d}\right)=E(10^{n+1}(\alpha-S_{n}))
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Periodicidad:
+\series default
+ Como 
+\begin_inset Formula $0\leq r_{n}<d\forall n$
+\end_inset
+
+, los 
+\begin_inset Formula $r_{n}$
+\end_inset
+
+ deben repetirse, es decir, 
+\begin_inset Formula $\exists m,q\in\mathbb{N},q>0:r_{m}=r_{m+q}$
+\end_inset
+
+.
+ Vemos por inducción que 
+\begin_inset Formula $a_{i}=a_{i+q}\forall i\geq m+1$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $i=m+1$
+\end_inset
+
+, 
+\begin_inset Formula $a_{m+1}$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{m+1}$
+\end_inset
+
+ son cociente y resto de 
+\begin_inset Formula $10r_{m}/d$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $a_{(m+q)+1}=a_{(m+1)+q}$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{(m+q)+1}=r_{(m+1)+q}$
+\end_inset
+
+ son cociente y resto de 
+\begin_inset Formula $10r_{m+q}/d=10r_{m}/d$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $a_{m+1}=a_{(m+1)+q}$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{m+1}=r_{(m+1)+q}$
+\end_inset
+
+.
+ El paso de inducción es análogo, partiendo de que 
+\begin_inset Formula $r_{i}=r_{i+q}$
+\end_inset
+
+ para obtener que 
+\begin_inset Formula $a_{i+1}=a_{(i+q)+1}$
+\end_inset
+
+ y 
+\begin_inset Formula $r_{i+1}=r_{(i+q)+1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Estructuras algebraicas
+\end_layout
+
+\begin_layout Standard
+Un conjunto 
+\begin_inset Formula $A\neq\emptyset$
+\end_inset
+
+ con una operación suma 
+\begin_inset Formula $+:A\times A\rightarrow A$
+\end_inset
+
+ es un 
+\series bold
+grupo abeliano
+\series default
+ si la suma es conmutativa, asociativa, existe un elemento neutro 
+\begin_inset Formula $0\in A$
+\end_inset
+
+ y todo 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ tiene opuesto (
+\begin_inset Formula $b\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $a+b=0$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Si además tiene una operación producto 
+\begin_inset Formula $\cdot:A\times A\rightarrow A$
+\end_inset
+
+, decimos que es un 
+\series bold
+anillo
+\series default
+ si con la suma es un grupo abeliano, el producto es asociativo, distribuye
+ a la suma y tiene neutro 
+\begin_inset Formula $1\in A$
+\end_inset
+
+.
+ Un anillo en que el producto es conmutativo es un 
+\series bold
+anillo conmutativo
+\series default
+, y si además todo 
+\begin_inset Formula $a\in A\backslash\{0\}$
+\end_inset
+
+ tiene inverso (
+\begin_inset Formula $b\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $ab=1$
+\end_inset
+
+), decimos que es un 
+\series bold
+cuerpo
+\series default
+.
+\end_layout
+
+\begin_layout Section
+Números reales
+\end_layout
+
+\begin_layout Standard
+Podemos construirlos partiendo de los racionales de 3 formas:
+\end_layout
+
+\begin_layout Enumerate
+Identificándolos con los desarrollos decimales infinitos.
+\end_layout
+
+\begin_layout Enumerate
+Mediante las 
+\series bold
+cortaduras de Dedekind
+\series default
+, conjuntos 
+\begin_inset Formula $\emptyset\neq\beta\subsetneq\mathbb{Q}$
+\end_inset
+
+ acotados superiormente y sin máximo tales que 
+\begin_inset Formula $y<x\in\beta\implies y\in\beta$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Considerando el conjunto cociente de cierta relación de equivalencia de
+ las sucesiones de Cauchy en 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Asumimos que es un conjunto no vacío 
+\begin_inset Formula $(\mathbb{R},+,\cdot)$
+\end_inset
+
+ que contiene a los racionales y satisface los siguientes axiomas:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Axiomas de cuerpo:
+\series default
+ 
+\begin_inset Formula $(\mathbb{R},+,\cdot)$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Axiomas de orden:
+\series default
+ 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ está totalmente ordenado, 
+\begin_inset Formula $x<y\implies x+z<y+z$
+\end_inset
+
+ y 
+\begin_inset Formula $x,y>0\implies xy>0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $x\in\mathbb{R}$
+\end_inset
+
+ es positivo si 
+\begin_inset Formula $x>0$
+\end_inset
+
+ y negativo si 
+\begin_inset Formula $x<0$
+\end_inset
+
+.
+ De aquí se tiene que si 
+\begin_inset Formula $x>0$
+\end_inset
+
+, su opuesto 
+\begin_inset Formula $-x<0$
+\end_inset
+
+, pues 
+\begin_inset Formula $x>0\implies x-x>0-x\implies0>-x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Axiomas de completitud:
+\series default
+ Todo subconjunto no vacío de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ acotado superiormente posee supremo.
+\end_layout
+
+\begin_layout Section
+Números complejos
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+números complejos
+\series default
+ al cuerpo definido por
+\begin_inset Formula 
+\[
+\mathbb{C}=\{(a,b)|a,b\in\mathbb{R}\}
+\]
+
+\end_inset
+
+junto con las operaciones 
+\begin_inset Formula $(a,b)+(c,d)=(a+c,b+d)$
+\end_inset
+
+ y 
+\begin_inset Formula $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$
+\end_inset
+
+.
+ Se representan en el plano cartesiano en las coordenadas 
+\begin_inset Formula $(a,b)$
+\end_inset
+
+.
+ Identificamos 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $\{(a,0)\}_{a\in\mathbb{R}}$
+\end_inset
+
+.
+ Definimos 
+\begin_inset Formula $i^{2}=-1$
+\end_inset
+
+ y escribimos 
+\begin_inset Formula $a+bi=(a,b)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $i^{n}=i^{m}\iff4|n-m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+conjugado
+\series default
+ de 
+\begin_inset Formula $z=a+bi\in\mathbb{C}$
+\end_inset
+
+ a 
+\begin_inset Formula $\overline{z}=a-bi$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overline{\overline{z}}=z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overline{z+w}=\overline{z}+\overline{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overline{z\cdot w}=\overline{z}\cdot\overline{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $z\neq0\implies\overline{z^{-1}}=\overline{z}^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $z\in\mathbb{R}\iff\overline{z}=z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $z=a+bi\in\mathbb{C}$
+\end_inset
+
+, su 
+\series bold
+parte real
+\series default
+ es 
+\begin_inset Formula $\text{Re}(z)=a$
+\end_inset
+
+, su 
+\series bold
+parte imaginaria
+\series default
+ es 
+\begin_inset Formula $\text{Im}(z)=b$
+\end_inset
+
+, su 
+\series bold
+módulo
+\series default
+ es 
+\begin_inset Formula $|z|=\sqrt{a^{2}+b^{2}}$
+\end_inset
+
+ y su 
+\series bold
+argumento
+\series default
+ es 
+\begin_inset Formula $\text{Arg}(z)=\theta=\arctan\frac{b}{a}$
+\end_inset
+
+, estableciendo primero el cuadrante de forma que 
+\begin_inset Formula $\cos(\theta)=\frac{a}{|z|}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sin(\theta)=\frac{b}{|z|}$
+\end_inset
+
+, y es único salvo múltiplos de 
+\begin_inset Formula $2\pi$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|z|^{2}=z\overline{z}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|z|=|\overline{z}|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|zw|=|z||w|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|z^{-1}|=|z|^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|\text{Re}(z)|\leq|z|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Desigualdad triangular:
+\series default
+ 
+\begin_inset Formula $|z+w|\leq|z|+|w|$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Como 
+\begin_inset Formula $z\overline{w}=\overline{\overline{z}w}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $z\overline{w}+\overline{z}w=2\text{Re}(z\overline{w})$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $|z+w|^{2}=(z+w)(\overline{z}+\overline{w})=z\overline{z}+w\overline{w}+z\overline{w}+\overline{z}w=|z|^{2}+|w|^{2}+2\text{Re}(z\overline{w})\leq|z|^{2}+|w|^{2}+2|z\overline{w}|=(|z|+|w|)^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $z=a+bi$
+\end_inset
+
+ con módulo 
+\begin_inset Formula $r$
+\end_inset
+
+ y argumento 
+\begin_inset Formula $\theta$
+\end_inset
+
+, la 
+\series bold
+representación polar
+\series default
+ de 
+\begin_inset Formula $z$
+\end_inset
+
+ es 
+\begin_inset Formula $z\mapsto(r,\theta)$
+\end_inset
+
+, pues 
+\begin_inset Formula $r$
+\end_inset
+
+ es la distancia al centro cartesiano y 
+\begin_inset Formula $\theta$
+\end_inset
+
+ el ángulo respecto del eje de abscisas.
+ Así, su 
+\series bold
+representación trigonométrica
+\series default
+ es 
+\begin_inset Formula $z\mapsto r(\cos\theta+i\sin\theta)$
+\end_inset
+
+, y si 
+\begin_inset Formula $z=(r,\theta)$
+\end_inset
+
+ y 
+\begin_inset Formula $w=(s,\sigma)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $zw=(rs,\theta+\sigma)$
+\end_inset
+
+.
+ De aquí se deduce el 
+\series bold
+teorema de De Moivre:
+\series default
+ Dado 
+\begin_inset Formula $z=(r,\theta)$
+\end_inset
+
+, 
+\begin_inset Formula $z^{n}=(r^{n},n\theta)$
+\end_inset
+
+.
+ Por tanto, si 
+\begin_inset Formula $z^{n}=(s,\alpha)$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $r=\sqrt[n]{s}$
+\end_inset
+
+ y 
+\begin_inset Formula $\theta=\frac{\alpha+2k\pi}{n},k\in\mathbb{Z}$
+\end_inset
+
+, con lo que todo número complejo tiene exactamente 
+\begin_inset Formula $n$
+\end_inset
+
+ raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas complejas.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, 
+\begin_inset Formula $\omega\in\mathbb{C}$
+\end_inset
+
+ es una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima de la unidad si 
+\begin_inset Formula $\omega^{n}=1$
+\end_inset
+
+, y es una 
+\series bold
+raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de la unidad
+\series default
+ si además 
+\begin_inset Formula $\omega^{m}\neq1$
+\end_inset
+
+ para 
+\begin_inset Formula $0<m<n$
+\end_inset
+
+.
+ Así, todo número complejo tiene
+\begin_inset Formula 
+\[
+\phi(n)=|\{m\in\{1,\dots,n-1\}:\text{mcd}(m,n)=1\}|
+\]
+
+\end_inset
+
+raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas primitivas.
+ Esta función se conoce como 
+\series bold
+función 
+\begin_inset Formula $\phi$
+\end_inset
+
+ de Euler
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Se tiene que
+\begin_inset Formula 
+\begin{eqnarray*}
+e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!} & \cos x=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!} & \sin x=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Por tanto tiene sentido definir que 
+\begin_inset Formula $e^{ip}=\cos p+i\sin p$
+\end_inset
+
+, pues 
+\begin_inset Formula $e^{ip}=1+ip+\frac{(ip)^{2}}{2!}+\dots=1+ip-\frac{p^{2}}{2!}-\frac{ip^{3}}{3!}+\frac{p^{4}}{4!}+\dots=\cos p+i\sin p$
+\end_inset
+
+.
+ Por tanto, podemos escribir 
+\begin_inset Formula $z=(r,\theta)\in\mathbb{C}$
+\end_inset
+
+ como 
+\begin_inset Formula $z=re^{\theta i}$
+\end_inset
+
+, y obtenemos la 
+\series bold
+identidad de Euler
+\series default
+:
+\begin_inset Formula 
+\[
+e^{\pi i}+1=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Conjuntos numerables y no numerables
+\end_layout
+
+\begin_layout Standard
+Un conjunto 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+a lo más numerable
+\series default
+ si 
+\begin_inset Formula $|A|\leq|\mathbb{N}|$
+\end_inset
+
+, 
+\series bold
+numerable
+\series default
+ si 
+\begin_inset Formula $|A|=|\mathbb{N}|$
+\end_inset
+
+ y 
+\series bold
+más que numerable
+\series default
+ si 
+\begin_inset Formula $|A|>|\mathbb{N}|$
+\end_inset
+
+.
+ 
+\series bold
+Teorema de Bernstein
+\series default
+ o 
+\series bold
+de Cantor-Schröeder-Bernstein (CSB):
+\series default
+ Dados dos conjuntos 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ tales que existen 
+\begin_inset Formula $f:A\rightarrow B$
+\end_inset
+
+ y 
+\begin_inset Formula $g:B\rightarrow A$
+\end_inset
+
+ inyectivas, entonces existe una biyección entre ellos.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $|\mathbb{N}|=|\mathbb{N}\times\mathbb{N}|$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Para simplificar, interpretamos 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ sin el 0.
+ Ordenamos las parejas de 
+\begin_inset Formula $\mathbb{N}\times\mathbb{N}$
+\end_inset
+
+ en orden lexicográfico y luego vamos contando en diagonal.
+ Entonces en cada diagonal de 
+\begin_inset Formula $(1,n)$
+\end_inset
+
+ a 
+\begin_inset Formula $(n,1)$
+\end_inset
+
+ están los pares cuyas coordenadas suman 
+\begin_inset Formula $n+1$
+\end_inset
+
+, y al terminar la diagonal habremos contado 
+\begin_inset Formula $S(n)=\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$
+\end_inset
+
+ pares.
+ Entonces 
+\begin_inset Formula $(1,n)\mapsto S(n-1)+1$
+\end_inset
+
+, 
+\begin_inset Formula $(2,n-1)\mapsto S(n-1)+2$
+\end_inset
+
+, etc.
+ Así, definimos 
+\begin_inset Formula $\varphi:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $\varphi(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$
+\end_inset
+
+ y vemos que es una biyección.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema:
+\series default
+ 
+\begin_inset Formula $|\mathbb{N}|=|\mathbb{Z}|=|\mathbb{Q}|<(0,1)=|\mathbb{R}|$
+\end_inset
+
+.
+ 
+\series bold
+Demostración
+\series default
+ de que 
+\begin_inset Formula $|\mathbb{N}|<(0,1)$
+\end_inset
+
+: La aplicación 
+\begin_inset Formula $f:\mathbb{N}\rightarrow(0,1)$
+\end_inset
+
+ con 
+\begin_inset Formula $f(n)=\frac{1}{n+1}$
+\end_inset
+
+ es inyectiva.
+ Para ver que no hay aplicaciones inyectivas 
+\begin_inset Formula $(0,1)\rightarrow\mathbb{N}$
+\end_inset
+
+ usamos el 
+\series bold
+método de la diagonal de Cantor
+\series default
+.
+ Supongamos que existe y hemos numerado todos los elementos en 
+\begin_inset Formula $(0,1)$
+\end_inset
+
+.
+ Si los escribimos en su forma decimal, tenemos
+\begin_inset Formula 
+\begin{eqnarray*}
+x_{1} & = & 0,x_{11}x_{12}x_{13}\cdots\\
+x_{2} & = & 0,x_{21}x_{22}x_{23}\cdots\\
+x_{3} & = & 0,x_{31}x_{32}x_{33}\cdots
+\end{eqnarray*}
+
+\end_inset
+
+etcétera.
+ Ahora, sea 
+\begin_inset Formula $(y_{n})_{n}$
+\end_inset
+
+ una secuencia de dígitos con 
+\begin_inset Formula $y_{n}\in\{0,\dots,9\}\backslash\{x_{nn}\}$
+\end_inset
+
+ e
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $y_{1}=\{1,\dots,8\}\backslash\{x_{nn}\}$
+\end_inset
+
+ (para evitar que el número formado sea 0 o 1).
+ Entonces este número difiere con cada uno de la lista en al menos un dígito.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n7.lyx b/cyn/n7.lyx
new file mode 100644
index 0000000..875b9a2
--- /dev/null
+++ b/cyn/n7.lyx
@@ -0,0 +1,2681 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Aritmética de los enteros
+\end_layout
+
+\begin_layout Standard
+Propiedades de 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Unicidad de los neutros:
+\series default
+ 
+\begin_inset Formula $\exists!0\in\mathbb{Z}:\forall a\in\mathbb{Z},0+a=a$
+\end_inset
+
+; 
+\begin_inset Formula $\exists!1\in\mathbb{Z}:\forall a\in\mathbb{Z},1a=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Unicidad de los opuestos:
+\series default
+ 
+\begin_inset Formula $\forall a\in\mathbb{Z},\exists!(-a)\in\mathbb{Z}:a+(-a)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Cancelación en sumas:
+\series default
+ 
+\begin_inset Formula $\forall a,b,c\in\mathbb{Z},(a+b=a+c\implies b=c)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Multiplicación por cero:
+\series default
+ 
+\begin_inset Formula $\forall a\in\mathbb{Z},a0=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Reglas de signos:
+\series default
+ 
+\begin_inset Formula $\forall a,b\in\mathbb{Z},(-(-a)=a\land a(-b)=(-a)b=-(ab)\land(-a)(-b)=ab)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Cancelación en productos:
+\series default
+ 
+\begin_inset Formula $\forall a,b,c\in\mathbb{Z},a\neq0,(ab=ac\implies b=c)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la división entera:
+\series default
+ 
+\begin_inset Formula $\forall a,b\in\mathbb{Z},\exists!q,r\in\mathbb{Z}:(a=bq+r\land0\leq r<|b|)$
+\end_inset
+
+.
+ Llamamos a 
+\begin_inset Formula $q$
+\end_inset
+
+ el 
+\series bold
+cociente
+\series default
+ de la división y a 
+\begin_inset Formula $r$
+\end_inset
+
+ el 
+\series bold
+resto
+\series default
+\SpecialChar endofsentence
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $a,b>0$
+\end_inset
+
+ y 
+\begin_inset Formula $R=\{x\in\mathbb{Z}|x\geq0\land\exists n\in\mathbb{Z}:x=a-bn\}\subseteq\mathbb{N}$
+\end_inset
+
+.
+ Sabemos que 
+\begin_inset Formula $R\neq\emptyset$
+\end_inset
+
+ porque 
+\begin_inset Formula $a=a-b\cdot0\in R$
+\end_inset
+
+.
+ Por tanto tiene primer elemento 
+\begin_inset Formula $r=a-bq\in R$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $r\geq b$
+\end_inset
+
+ entonces 
+\begin_inset Formula $0\leq r-b\in R\#$
+\end_inset
+
+, luego 
+\begin_inset Formula $r<b$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a<0$
+\end_inset
+
+ y 
+\begin_inset Formula $b>0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $-a>0$
+\end_inset
+
+ y 
+\begin_inset Formula $-a=bq+r$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r<b$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $r=0$
+\end_inset
+
+, 
+\begin_inset Formula $a=b(-q)+0$
+\end_inset
+
+, y si 
+\begin_inset Formula $r\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $a=b(-q)-r=b(-q-1)+(b-r)$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $b<0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $-b>0$
+\end_inset
+
+ y 
+\begin_inset Formula $a=(-b)q+r$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r<-b=|b|$
+\end_inset
+
+, luego 
+\begin_inset Formula $a=b(-q)+r$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r<|b|$
+\end_inset
+
+.
+ Finalmente, si 
+\begin_inset Formula $a=0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $0=b\cdot0+0$
+\end_inset
+
+.
+ Para la unicidad de 
+\begin_inset Formula $q$
+\end_inset
+
+ y 
+\begin_inset Formula $r$
+\end_inset
+
+, supongamos 
+\begin_inset Formula $a=bq+r=bq'+r'$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r,r'<|b|$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $b(q-q')=r-r'$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $|b||q-q'|=|r-r'|$
+\end_inset
+
+, pero como 
+\begin_inset Formula $0\leq r,r'<|b|$
+\end_inset
+
+, entonces 
+\begin_inset Formula $q-q'=0$
+\end_inset
+
+ y 
+\begin_inset Formula $r-r'=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Decimos que 
+\series bold
+
+\begin_inset Formula $b$
+\end_inset
+
+ divide a 
+\begin_inset Formula $a$
+\end_inset
+
+
+\series default
+ o que 
+\series bold
+
+\begin_inset Formula $a$
+\end_inset
+
+ es múltiplo de 
+\begin_inset Formula $b$
+\end_inset
+
+
+\series default
+ (
+\begin_inset Formula $b|a$
+\end_inset
+
+) si 
+\begin_inset Formula $\exists c\in\mathbb{Z}:a=bc$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a\neq0$
+\end_inset
+
+, también decimos que 
+\series bold
+
+\begin_inset Formula $b$
+\end_inset
+
+ es divisor de 
+\begin_inset Formula $a$
+\end_inset
+
+
+\series default
+.
+ Para 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $b|a$
+\end_inset
+
+ equivale a que la división entera de 
+\begin_inset Formula $a$
+\end_inset
+
+ entre 
+\begin_inset Formula $b$
+\end_inset
+
+ dé resto 0.
+\end_layout
+
+\begin_layout Enumerate
+La divisibilidad es reflexiva y transitiva.
+\end_layout
+
+\begin_layout Enumerate
+No es antisimétrica, pero 
+\begin_inset Formula $a|b\land b|a\implies|a|=|b|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|b\iff a|-b$
+\end_inset
+
+, con lo que si 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $b$
+\end_inset
+
+ y 
+\begin_inset Formula $-b$
+\end_inset
+
+ tienen los mismos divisores.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|b\iff-a|b$
+\end_inset
+
+, luego 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $-a$
+\end_inset
+
+ tienen los mismos múltiplos.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c|a\land c|b\implies c|ra+sb$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|b\land c|d\implies ac|bd$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|b\implies ca|cb$
+\end_inset
+
+.
+ El recíproco es cierto si 
+\begin_inset Formula $c\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $a|b\implies|a|\leq|b|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+, su 
+\series bold
+máximo común divisor
+\series default
+ es 
+\begin_inset Formula $\text{mcd}(a,b)=\max\{d\in\mathbb{Z}:d|a\land d|b\}$
+\end_inset
+
+ (excepción: 
+\begin_inset Formula $\text{mcd}(0,0)=0$
+\end_inset
+
+).
+ Este existe porque el conjunto de divisores comunes es no vacío (contiene
+ al 1) y finito, luego tiene máximo.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcd}(a,b)=\text{mcd}(a,|b|)=\text{mcd}(|a|,|b|)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcd}(a,0)=|a|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcd}(a,b)=0\iff a=b=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+ con alguno distinto de 0, 
+\begin_inset Formula $\text{mcd}(a,b)=\min\{ra+sb>0|r,s\in\mathbb{Z}\}$
+\end_inset
+
+, y todo divisor común de 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ lo es de 
+\begin_inset Formula $\text{mcd}(a,b)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Dado 
+\begin_inset Formula $\emptyset\neq D=\{ra+sb>0|r,s\in\mathbb{Z}\}\subseteq\mathbb{Z}^{+}$
+\end_inset
+
+, existe 
+\begin_inset Formula $\delta=\min D$
+\end_inset
+
+.
+ Existen entonces 
+\begin_inset Formula $\alpha,\beta\in\mathbb{Z}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\delta=\alpha a+\beta b$
+\end_inset
+
+.
+ Por el algoritmo de la división, 
+\begin_inset Formula $a=\delta q+r$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r<\delta$
+\end_inset
+
+, luego 
+\begin_inset Formula $r=(1-\alpha q)a+(-q\beta)b$
+\end_inset
+
+, luego 
+\begin_inset Formula $r$
+\end_inset
+
+ es combinación lineal y entonces 
+\begin_inset Formula $r\in D$
+\end_inset
+
+ o 
+\begin_inset Formula $r=0$
+\end_inset
+
+.
+ Lo primero es imposible porque 
+\begin_inset Formula $r<\delta=\min D$
+\end_inset
+
+, luego 
+\begin_inset Formula $r=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\delta|a$
+\end_inset
+
+.
+ Análogamente 
+\begin_inset Formula $\delta|b$
+\end_inset
+
+.
+ Que sea máximo, y que todo divisor común de 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ lo sean de 
+\begin_inset Formula $\delta$
+\end_inset
+
+, se desprende de que 
+\begin_inset Formula $c|a\land c|b\implies c|\alpha a+\beta b=\delta$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí que para todo 
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+ existen 
+\begin_inset Formula $r,s\in\mathbb{Z}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\text{mcd}(a,b)=ra+sb$
+\end_inset
+
+.
+ Una expresión de la forma 
+\begin_inset Formula $d=ra+sb$
+\end_inset
+
+ es una 
+\series bold
+identidad de Bézout
+\series default
+.
+ En particular, si 
+\begin_inset Formula $a=da'$
+\end_inset
+
+ y 
+\begin_inset Formula $b=db'$
+\end_inset
+
+ con 
+\begin_inset Formula $d=\text{mcd}(a,b)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{mcd}(a',b')=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $d=\text{mcd}(a,b)$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $d|a$
+\end_inset
+
+, 
+\begin_inset Formula $d|b$
+\end_inset
+
+, 
+\begin_inset Formula $c|a\land c|b\implies c|d$
+\end_inset
+
+ y 
+\begin_inset Formula $d\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Las propiedades (1) y (3) son por definición, y la (2) la acabamos de demostrar.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $a\neq0$
+\end_inset
+
+ o 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $d$
+\end_inset
+
+ es el mayor entero que divide a 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a=b=0$
+\end_inset
+
+, como 
+\begin_inset Formula $0|a,b$
+\end_inset
+
+, entonces 
+\begin_inset Formula $0|d$
+\end_inset
+
+, luego 
+\begin_inset Formula $d=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El máximo común divisor de 
+\begin_inset Formula $a_{1},\dots,a_{n}$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{mcd}(a_{1},\dots,a_{n})=\max\{d\in\mathbb{Z}:\forall i,d|a_{i}\}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{mcd}(a_{1},\dots,a_{n})=\text{mcd}(\text{mcd}(a_{1},a_{2}),a_{3},\dots,a_{n})$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $d:=\text{mcd}(a_{1},\dots,a_{n})$
+\end_inset
+
+, como 
+\begin_inset Formula $d||a_{1},\dots,a_{n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $d|(f:=\text{mcd}(a_{1},a_{2})),a_{3},\dots,a_{n}|e:=\text{mcd}(\text{mcd}(a_{1},a_{2}),a_{3},\dots,a_{n})$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $d|e$
+\end_inset
+
+.
+ Pero 
+\begin_inset Formula $e|f,a_{3},\dots,a_{n}$
+\end_inset
+
+, luego 
+\begin_inset Formula $e|a_{1},\dots,a_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $e|d$
+\end_inset
+
+, y como 
+\begin_inset Formula $d,e\geq0$
+\end_inset
+
+, 
+\begin_inset Formula $d=e$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema:
+\series default
+ Dados 
+\begin_inset Formula $a_{1},\dots,a_{n}\in\mathbb{Z}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\text{mcd}(a_{1},\dots,a_{n})=\min\left\{ \sum_{i=1}^{n}r_{i}a_{i}>0|r_{i}\in\mathbb{Z}\right\} $
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $d=\text{mcd}(a_{1},\dots,a_{n})$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $d|a_{1},\dots,a_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $c|a_{1},\dots,a_{n}\implies c|d$
+\end_inset
+
+ y 
+\begin_inset Formula $d\geq0$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+ son 
+\series bold
+coprimos
+\series default
+ o 
+\series bold
+primos entre sí
+\series default
+ si 
+\begin_inset Formula $\text{mcd}(a,b)=1$
+\end_inset
+
+, es decir, si 
+\begin_inset Formula $\exists\alpha,\beta\in\mathbb{Z}:\alpha a+\beta b=1$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ son coprimos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|bc\implies a|c$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula $1=\alpha a+\beta b$
+\end_inset
+
+, multiplicando por 
+\begin_inset Formula $c$
+\end_inset
+
+, 
+\begin_inset Formula $c=\alpha ac+\beta bc$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $a|bc$
+\end_inset
+
+, 
+\begin_inset Formula $c=\alpha ca+\beta na$
+\end_inset
+
+ y 
+\begin_inset Formula $a|c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|c\land b|c\implies ab|c$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{multline*}
+\begin{array}{c}
+1=ra+sb\\
+\frac{c}{a},\frac{c}{b}\in\mathbb{Z}
+\end{array}\implies\frac{c}{a}=\frac{c}{a}ra+\frac{c}{a}sb=\frac{c}{b}rb+\frac{c}{a}sb=b\left(\frac{c}{b}r+\frac{c}{a}s\right)\implies\\
+\implies c=ab\left(\frac{c}{b}r+\frac{c}{a}s\right)\implies ab|c
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Se tiene que 
+\begin_inset Formula $\text{mcd}(a,b)=\text{mcd}(a-sb,b)=\text{mcd}(a,b-sa)$
+\end_inset
+
+, y en particular, si 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $a=bq+r$
+\end_inset
+
+ con 
+\begin_inset Formula $0\leq r<b$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{mcd}(a,b)=\text{mcd}(b,r)$
+\end_inset
+
+.
+ La aplicación repetida de lo anterior se conoce como 
+\series bold
+algoritmo de Euclides
+\series default
+.
+ También permite obtener identidades de Bézout.
+ Si llamamos 
+\begin_inset Formula $(a,b)=\text{mcd}(a,b)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{eqnarray*}
+a=bq_{1}+r_{1} & (a,b)=(b,r_{1}) & r_{1}<b\\
+b=r_{1}q_{2}+r_{2} & (b,r_{1})=(r_{1},r_{2}) & r_{2}<r_{1}\\
+ & \vdots\\
+r_{n-2}=r_{n-1}q_{n}+0 & (r_{n-2},r_{n-1})=(r_{n-1},0)=r_{n-1} & 0<r_{n-1}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\begin_inset Formula $b>r_{1}>\dots\geq0$
+\end_inset
+
+, el algoritmo acaba en un número finito de pasos.
+ Además, cada dos pasos del algoritmo, el resto se reduce a la mitad.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $a=bq+r$
+\end_inset
+
+, 
+\begin_inset Formula $b=rq'+s$
+\end_inset
+
+ y 
+\begin_inset Formula $r=sq''+t$
+\end_inset
+
+, si 
+\begin_inset Formula $s\leq\frac{1}{2}r$
+\end_inset
+
+ entonces 
+\begin_inset Formula $t<s\leq\frac{1}{2}r$
+\end_inset
+
+ y hemos terminado, y si 
+\begin_inset Formula $s>\frac{1}{2}r$
+\end_inset
+
+, entonces 
+\begin_inset Formula $q''=1$
+\end_inset
+
+ y 
+\begin_inset Formula $t=r-s<r-\frac{1}{2}r=\frac{1}{2}r$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a,b\in\mathbb{Z}^{*}$
+\end_inset
+
+, su 
+\series bold
+mínimo común múltiplo
+\series default
+ es 
+\begin_inset Formula $\text{mcm}(a,b)=\min\{m\in\mathbb{Z}^{+}:a|m\land b|m\}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a$
+\end_inset
+
+ o 
+\begin_inset Formula $b$
+\end_inset
+
+ son 0, entonces 
+\begin_inset Formula $\text{mcm}(a,b)=0$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcm}(a,b)=\text{mcm}(a,|b|)=\text{mcm}(|a|,|b|)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcm}(a,b)=0\iff a=0\lor b=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcm}(a,ab)=|ab|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Teorema:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{mcm}(a,b)\text{mcd}(a,b)=|ab|$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Para 
+\begin_inset Formula $a,b>0$
+\end_inset
+
+, sea 
+\begin_inset Formula $d=\text{mcd}(a,b)$
+\end_inset
+
+ con 
+\begin_inset Formula $a=da'$
+\end_inset
+
+ y 
+\begin_inset Formula $b=db'$
+\end_inset
+
+, sea 
+\begin_inset Formula $m=a'b'd$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $a|m$
+\end_inset
+
+ y 
+\begin_inset Formula $b|m$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $c=\alpha a=\beta b$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha,\beta\in\mathbb{Z}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\alpha da'=\beta db'$
+\end_inset
+
+, luego 
+\begin_inset Formula $\alpha a'=\beta b'$
+\end_inset
+
+, y como 
+\begin_inset Formula $a'$
+\end_inset
+
+ y 
+\begin_inset Formula $b'$
+\end_inset
+
+ son coprimos, 
+\begin_inset Formula $a'|\beta$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta=\gamma a'$
+\end_inset
+
+ con 
+\begin_inset Formula $\gamma\in\mathbb{Z}$
+\end_inset
+
+.
+ Sustituyendo, 
+\begin_inset Formula $c=\gamma a'b=\gamma a'db'=\gamma m\geq m$
+\end_inset
+
+, luego 
+\begin_inset Formula $m=\text{mcm}(a,b)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a|c\land b|c\implies\text{mcm}(a,b)|c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El mínimo común múltiplo de 
+\begin_inset Formula $a_{1},\dots,a_{n}$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{mcm}(a_{1},\dots,a_{n})=\min\{m\in\mathbb{Z}^{+}:\forall i,a_{i}|m\}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $m=\text{mcm}(a_{1},\dots,a_{n})$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $a_{1},\dots,a_{n}|m$
+\end_inset
+
+, 
+\begin_inset Formula $a_{1},\dots,a_{n}|c\implies m|c$
+\end_inset
+
+ y 
+\begin_inset Formula $m\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una ecuación del tipo 
+\begin_inset Formula $ax+by=c$
+\end_inset
+
+ en la que se buscan soluciones enteras es una 
+\series bold
+ecuación diofántica lineal
+\series default
+, en este caso de dos variables.
+ Tiene solución si y sólo si 
+\begin_inset Formula $d=\text{mcd}(a,b)|c$
+\end_inset
+
+, y entonces estas son de la forma
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{ccc}
+x & = & x_{0}+x'\\
+y & = & y_{0}+y'
+\end{array}\right.
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $x_{0},y_{0}$
+\end_inset
+
+ es una solución particular y 
+\begin_inset Formula $x',y'$
+\end_inset
+
+ es una solución de la 
+\series bold
+ecuación homogénea asociada
+\series default
+, 
+\begin_inset Formula $ax+by=0$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $\alpha a+\beta b=d$
+\end_inset
+
+ y 
+\begin_inset Formula $c=c'd$
+\end_inset
+
+, entonces 
+\begin_inset Formula $x_{0}=c'\alpha$
+\end_inset
+
+ e 
+\begin_inset Formula $y_{0}=c'\beta$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $x,y\in\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $ax+by=c$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $d|ax+by=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Multiplicando la identidad de Bézout, 
+\begin_inset Formula $(c'\alpha)a+(c'\beta)b=c'd=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $d=\text{mcd}(a,b)$
+\end_inset
+
+, 
+\begin_inset Formula $a=a'd$
+\end_inset
+
+ y 
+\begin_inset Formula $b=b'd$
+\end_inset
+
+, las soluciones de 
+\begin_inset Formula $ax+by=0$
+\end_inset
+
+ son
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{ccc}
+x & = & -b't\\
+y & = & a't
+\end{array}\right.
+\]
+
+\end_inset
+
+para cualquier 
+\begin_inset Formula $t\in\mathbb{Z}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $ax=-by$
+\end_inset
+
+, luego 
+\begin_inset Formula $a'x=-b'y$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $a'$
+\end_inset
+
+ y 
+\begin_inset Formula $b'$
+\end_inset
+
+ son coprimos y 
+\begin_inset Formula $a'|-b'y$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a'|y$
+\end_inset
+
+, luego existe 
+\begin_inset Formula $t\in\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $y=a't$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $a'x=-b'a't$
+\end_inset
+
+ y 
+\begin_inset Formula $x=-b't$
+\end_inset
+
+.
+ Multiplicando, todos los enteros de esta forma son solución.
+\end_layout
+
+\begin_layout Standard
+Un entero 
+\begin_inset Formula $p\neq1,-1$
+\end_inset
+
+ es 
+\series bold
+primo
+\series default
+ si sus únicos divisores son 1, 
+\begin_inset Formula $-1$
+\end_inset
+
+, 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $-p$
+\end_inset
+
+.
+ Así,
+\begin_inset Formula 
+\[
+p\text{ es primo}\iff(p|ab\implies p|a\lor p|b)\iff(p|a_{1}\cdots a_{n}\implies\exists i:p|a_{i})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Si 
+\begin_inset Formula $p|a$
+\end_inset
+
+ ya está.
+ Si no, 
+\begin_inset Formula $\text{mcd}(p,a)=1$
+\end_inset
+
+ y 
+\begin_inset Formula $p|b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Por inducción con 
+\begin_inset Formula $a_{1}\cdots a_{n}=a_{1}(a_{2}\cdots a_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Si 
+\begin_inset Formula $a|p$
+\end_inset
+
+ entonces 
+\begin_inset Formula $p=ab$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $b$
+\end_inset
+
+, y bien 
+\begin_inset Formula $p|a$
+\end_inset
+
+ (con lo que 
+\begin_inset Formula $a=p$
+\end_inset
+
+ o 
+\begin_inset Formula $a=-p$
+\end_inset
+
+) o 
+\begin_inset Formula $p|b$
+\end_inset
+
+ (con lo que 
+\begin_inset Formula $a=1$
+\end_inset
+
+ o 
+\begin_inset Formula $a=-1$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema Fundamental de la Aritmética:
+\series default
+ Todo entero distinto de 
+\begin_inset Formula $0$
+\end_inset
+
+ y 
+\begin_inset Formula $\pm1$
+\end_inset
+
+ puede escribirse como producto de primos, y la factorización es única salvo
+ signo y orden.
+ 
+\series bold
+Demostración:
+\series default
+ Consideremos el conjunto de todos los positivos distintos de 1 que no se
+ factorizan en primos y, si este no es vacío, sea 
+\begin_inset Formula $a\in\mathbb{Z}$
+\end_inset
+
+ su mínimo.
+ 
+\begin_inset Formula $a$
+\end_inset
+
+ no es primo, luego 
+\begin_inset Formula $a=bc$
+\end_inset
+
+ con 
+\begin_inset Formula $b,c\in\mathbb{Z}^{+}\backslash\{1\}$
+\end_inset
+
+.
+ Pero como 
+\begin_inset Formula $a$
+\end_inset
+
+ es mínimo, entonces 
+\begin_inset Formula $b$
+\end_inset
+
+ y 
+\begin_inset Formula $c$
+\end_inset
+
+ sí se factorizan en primos, luego 
+\begin_inset Formula $a$
+\end_inset
+
+ también.
+\begin_inset Formula $\#$
+\end_inset
+
+ Ahora sea 
+\begin_inset Formula $a=p_{1}\cdots p_{n}=q_{1}\cdots q_{m}$
+\end_inset
+
+ con 
+\begin_inset Formula $p_{1}\cdots p_{n},q_{1}\cdots q_{m}$
+\end_inset
+
+ primos y supongamos 
+\begin_inset Formula $n\leq m$
+\end_inset
+
+.
+ Procedemos por inducción sobre 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n=1$
+\end_inset
+
+, 
+\begin_inset Formula $a=p_{1}=q_{1}\cdots q_{m}$
+\end_inset
+
+, y como 
+\begin_inset Formula $p_{1}$
+\end_inset
+
+ no tiene más divisores primos que 
+\begin_inset Formula $-p_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $p_{1}$
+\end_inset
+
+, debe ser 
+\begin_inset Formula $m=1$
+\end_inset
+
+ y 
+\begin_inset Formula $q_{1}=p_{1}$
+\end_inset
+
+.
+ Si suponemos el resultado válido para 
+\begin_inset Formula $n-1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $p_{n}$
+\end_inset
+
+ divide a 
+\begin_inset Formula $a=q_{1}\cdots q_{n}$
+\end_inset
+
+ y por tanto divide a algún 
+\begin_inset Formula $i\in\{1,\dots,m\}$
+\end_inset
+
+.
+ Reordenamos los factores para obtener 
+\begin_inset Formula $i=m$
+\end_inset
+
+, es decir 
+\begin_inset Formula $p_{n}|q_{m}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $q_{m}=\pm p_{n}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $p_{1}\cdots p_{n-1}p_{n}=q_{1}\cdots q_{m-1}(\pm p_{n})$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $p_{1}\cdots p_{n-1}=\pm q_{1}\cdots q_{m-1}$
+\end_inset
+
+ y 
+\begin_inset Formula $n-1=m-1$
+\end_inset
+
+, luego 
+\begin_inset Formula $n=m$
+\end_inset
+
+ y además, después de ordenar si hiciera falta, 
+\begin_inset Formula $q_{i}=\pm p_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, para 
+\begin_inset Formula $a\in\mathbb{Z},a\neq0,\pm1$
+\end_inset
+
+, 
+\begin_inset Formula $a=\pm p_{1}^{n_{1}}\cdots p_{s}^{n_{s}}$
+\end_inset
+
+ y estos primos y sus exponentes son únicos (salvo orden).
+ Entonces podemos calcular el 
+\begin_inset Formula $\text{mcd}(a,b)$
+\end_inset
+
+ tomando el producto de primos comunes a 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ elevados a la mínima potencia y el 
+\begin_inset Formula $\text{mcm}(a,b)$
+\end_inset
+
+ tomando el producto de primos entre ambos elevados a la máxima potencia.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, el conjunto de los números primos es infinito.
+ Si no lo fuera, y fuera 
+\begin_inset Formula $\{p_{1},\dots,p_{n}\}$
+\end_inset
+
+, el número 
+\begin_inset Formula $N:=p_{1}\cdots p_{n}+1$
+\end_inset
+
+ también lo es.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Congruencias
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $x,y\in\mathbb{Z},m\in\mathbb{Z}^{+}$
+\end_inset
+
+, 
+\begin_inset Formula $x$
+\end_inset
+
+ e 
+\begin_inset Formula $y$
+\end_inset
+
+ son 
+\series bold
+congruentes módulo 
+\begin_inset Formula $m$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $x\equiv y\mod m$
+\end_inset
+
+ ó 
+\begin_inset Formula $x\equiv y\,(m)$
+\end_inset
+
+, si 
+\begin_inset Formula $m|x-y$
+\end_inset
+
+.
+ Esta relación es de equivalencia.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $r$
+\end_inset
+
+ es el resto de 
+\begin_inset Formula $a/m$
+\end_inset
+
+ entonces 
+\begin_inset Formula $a\equiv r\,(m)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\equiv b\,(m)\land0\leq a,b<m\implies a=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\equiv b\,(m)$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ dan el mismo resto entre 
+\begin_inset Formula $m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\equiv a'\,(m)\land b\equiv b'\,(m)\implies a+b\equiv a'+b'\,(m)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $a-a'=\lambda m$
+\end_inset
+
+ y 
+\begin_inset Formula $b-b'=\mu m$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\lambda,\mu\in\mathbb{Z}$
+\end_inset
+
+, luego 
+\begin_inset Formula $(a+b)-(a'+b')=(\lambda+\mu)m$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\equiv a'\,(m)\land b\equiv b'\,(m)\implies ab\equiv a'b'\,(m)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula 
+\[
+ab-a'b'=(a'+\lambda m)(b'+\mu m)-a'b'=a'b'+(a'\mu+b'\lambda+\lambda\mu m)m-a'b'\equiv0\,(m)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\equiv b\,(m)\implies ac\equiv bc\,(m)$
+\end_inset
+
+.
+ El recíproco es cierto si 
+\begin_inset Formula $c$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+ son coprimos.
+\begin_inset Newline newline
+\end_inset
+
+La primera parte se sigue de lo anterior.
+ Para la segunda, 
+\begin_inset Formula $m|ac-bc=(a-b)c\implies m|a-b\implies a\equiv b\,(m)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c\neq0\implies(a\equiv b\,(m)\iff ac\equiv bc\,(mc))$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $a-b=\lambda m\iff ac-bc=\lambda mc$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Denotamos la clase de equivalencia (llamada 
+\series bold
+clase de congruencia módulo 
+\begin_inset Formula $m$
+\end_inset
+
+
+\series default
+) de 
+\begin_inset Formula $a\in\mathbb{Z}$
+\end_inset
+
+ por 
+\begin_inset Formula $\overline{a}$
+\end_inset
+
+, y su 
+\series bold
+representante canónico
+\series default
+ es el resto de 
+\begin_inset Formula $a/m$
+\end_inset
+
+.
+ Llamamos entonces 
+\begin_inset Formula $\mathbb{Z}/(m)$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{Z}_{m}$
+\end_inset
+
+ al conjunto cociente, que tiene exactamente 
+\begin_inset Formula $m$
+\end_inset
+
+ elementos.
+ Así, para 
+\begin_inset Formula $\overline{a},\overline{b}\in\mathbb{Z}_{m}$
+\end_inset
+
+, definimos 
+\begin_inset Formula $\overline{a}+\overline{b}=\overline{a+b}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{a}\cdot\overline{b}=\overline{a\cdot b}$
+\end_inset
+
+, y vemos que están bien definidas y que 
+\begin_inset Formula $\mathbb{Z}_{m}$
+\end_inset
+
+ es un anillo conmutativo.
+ Dado 
+\begin_inset Formula $\overline{a}\in\mathbb{Z}_{m}$
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{multline*}
+\overline{a}\text{ tiene inverso (}\exists\overline{b}\in\mathbb{Z}_{m}:\overline{a}\cdot\overline{b}=1\text{)}\iff\overline{a}\text{ es cancelable (}\overline{a}\cdot\overline{x}=\overline{a}\cdot\overline{y}\implies\overline{x}=\overline{y}\text{)}\iff\\
+\iff\text{mcd}(a,m)=1
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Multiplicando por 
+\begin_inset Formula $\overline{a}^{-1}$
+\end_inset
+
+ en ambos lados de 
+\begin_inset Formula $\overline{a}\cdot\overline{x}=\overline{a}\cdot\overline{y}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Si 
+\begin_inset Formula $\text{mcd}(a,m)=d>1$
+\end_inset
+
+, sean 
+\begin_inset Formula $a=a'd$
+\end_inset
+
+ y 
+\begin_inset Formula $m=m'd$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overline{a}\cdot\overline{m'}=\overline{a'}\cdot\overline{d}\cdot\overline{m'}=\overline{a'}\cdot\overline{m}=\overline{0}=\overline{a}\cdot\overline{0}$
+\end_inset
+
+, pero 
+\begin_inset Formula $\overline{m'}\neq\overline{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Existen 
+\begin_inset Formula $r,s\in\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $ra+sm=1$
+\end_inset
+
+, luego 
+\begin_inset Formula $\overline{r}\cdot\overline{a}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, 
+\begin_inset Formula $\mathbb{Z}_{m}$
+\end_inset
+
+ es un cuerpo si y sólo si 
+\begin_inset Formula $m$
+\end_inset
+
+ es primo, pues entonces todos los elementos tienen inverso.
+\end_layout
+
+\begin_layout Standard
+Un entero es divisible por 3 si y sólo si la suma de sus cifras lo es.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $3|m\iff m\equiv0\,(3)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $10\equiv1\,(3)$
+\end_inset
+
+, luego 
+\begin_inset Formula $10^{s}\equiv1\,(3)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $s$
+\end_inset
+
+ y si 
+\begin_inset Formula $m$
+\end_inset
+
+ se escribe como 
+\begin_inset Formula $a_{n}\cdots a_{0}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $m=a_{n}10^{n}+\dots+a_{0}\equiv a_{n}+\dots+a_{0}\,(3)$
+\end_inset
+
+.
+ De forma parecida se pueden sacar reglas para el 9 y el 11.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a,b,t\in\mathbb{Z}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\overline{t}\text{ es sol. de }\overline{a}x=\overline{b}\in\mathbb{Z}_{m}\iff t\text{ es sol. de }ax\equiv b\,(m)\iff\exists s\in\mathbb{Z}:(t,s)\text{ es sol. de }ax-my=b
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La ecuación 
+\begin_inset Formula $ax\equiv b\,(m)$
+\end_inset
+
+ tiene solución si y sólo si 
+\begin_inset Formula $d:=\text{mcd}(a,m)|b$
+\end_inset
+
+, y las soluciones son todos los enteros 
+\begin_inset Formula $x=x_{0}+\lambda\frac{m}{d}$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda\in\mathbb{Z}$
+\end_inset
+
+, donde 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ es una solución particular, de modo que la ecuación tiene 
+\begin_inset Formula $d$
+\end_inset
+
+ soluciones distintas módulo 
+\begin_inset Formula $m$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $ax\equiv b$
+\end_inset
+
+ equivale a la ecuación diofántica 
+\begin_inset Formula $ax-my=b$
+\end_inset
+
+, que tiene solución si y sólo si 
+\begin_inset Formula $d|b$
+\end_inset
+
+.
+ Sean pues 
+\begin_inset Formula $b=db'$
+\end_inset
+
+, 
+\begin_inset Formula $a=da'$
+\end_inset
+
+ y 
+\begin_inset Formula $m=dm'$
+\end_inset
+
+, entonces 
+\begin_inset Formula $ax-my=b$
+\end_inset
+
+ equivale a 
+\begin_inset Formula $a'x-m'y=b'$
+\end_inset
+
+ y las soluciones son
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{ccc}
+x & = & x_{0}+m'\lambda\\
+y & = & y_{0}+a'\lambda
+\end{array}\right.
+\]
+
+\end_inset
+
+Entonces 
+\begin_inset Formula $x_{0}+\lambda m'\equiv x_{0}+\mu m'\,(m)\iff\lambda m'\equiv\mu m'\,(dm')\iff\lambda\equiv\mu\,(d)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Teorema Chino de los Restos
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $b_{1},\dots,b_{k}\in\mathbb{Z}$
+\end_inset
+
+ arbitrarios y 
+\begin_inset Formula $m_{1},\dots,m_{k}\in\mathbb{Z}^{+}$
+\end_inset
+
+ coprimos dos a dos, el sistema de congruencias
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{cc}
+x\equiv b_{1} & (m_{1})\\
+\vdots\\
+x\equiv b_{k} & (m_{k})
+\end{array}\right.
+\]
+
+\end_inset
+
+tiene solución única módulo 
+\begin_inset Formula $M:=m_{1}\cdots m_{k}$
+\end_inset
+
+.
+ En particular, esta es 
+\begin_inset Formula $b_{1}M_{1}N_{1}+\dots+b_{k}M_{k}N_{k}$
+\end_inset
+
+, donde 
+\begin_inset Formula $M_{i}=\frac{M}{M_{i}}$
+\end_inset
+
+ y 
+\begin_inset Formula $N_{i}$
+\end_inset
+
+ es tal que 
+\begin_inset Formula $M_{i}N_{i}\equiv1\,(m_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es un número primo que divide a 
+\begin_inset Formula $M_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+, entonces divide a algún 
+\begin_inset Formula $m_{j}$
+\end_inset
+
+ con 
+\begin_inset Formula $j\neq i$
+\end_inset
+
+, lo cual contradice que 
+\begin_inset Formula $\text{mcd}(m_{i},m_{j})=1$
+\end_inset
+
+, luego 
+\begin_inset Formula $M_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+ son coprimos y 
+\begin_inset Formula $M_{i}$
+\end_inset
+
+ tiene inverso 
+\begin_inset Formula $N_{i}$
+\end_inset
+
+ módulo 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+, teniendo en cuenta que 
+\begin_inset Formula $M_{i}N_{i}\equiv0\,(m_{j})$
+\end_inset
+
+ para 
+\begin_inset Formula $j\neq i$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x_{0}=b_{1}M_{1}N_{1}+\dots+b_{k}M_{k}N_{k}$
+\end_inset
+
+ es solución del sistema.
+ Ahora bien, si 
+\begin_inset Formula $x$
+\end_inset
+
+ e 
+\begin_inset Formula $y$
+\end_inset
+
+ son soluciones del sistema, 
+\begin_inset Formula $x,y\equiv b_{i}\,(m_{i})$
+\end_inset
+
+, luego 
+\begin_inset Formula $x\equiv y\,(m_{i})$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $x-y$
+\end_inset
+
+ es múltiplo de todos los 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $x\equiv y\,(M)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si los módulos no son coprimos, intentamos simplificar cada ecuación dividiéndol
+a entre un número, pues 
+\begin_inset Formula $a'dx\equiv b'd\,(m'd)\iff a'x\equiv b'\,(m')$
+\end_inset
+
+.
+ Si esto no es posible, resolvemos una ecuación y sustituimos en el resto.
+\end_layout
+
+\begin_layout Section
+Teoremas de Euler y Fermat
+\end_layout
+
+\begin_layout Standard
+Denotamos 
+\begin_inset Formula $\mathbb{Z}_{m}^{*}=\{x\in\mathbb{Z}_{m}|x\text{ es invertible}\}$
+\end_inset
+
+, y definimos la 
+\series bold
+función 
+\begin_inset Formula $\phi$
+\end_inset
+
+ de Euler
+\series default
+ como 
+\begin_inset Formula $\phi:\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi(m)=|\{x\in\mathbb{N}|1\leq x\leq m\land\text{mcd}(x,m)=1\}|=|\mathbb{Z}_{m}^{*}|$
+\end_inset
+
+.
+ Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\phi(p)=p-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\phi(p^{n})=p^{n-1}(p-1)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Los no-coprimos con 
+\begin_inset Formula $p^{n}$
+\end_inset
+
+ son precisamente los múltiplos de 
+\begin_inset Formula $p$
+\end_inset
+
+, por lo que estos son 
+\begin_inset Formula $\frac{p^{n}}{p}=p^{n-1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(p^{n})=p^{n}-p^{n-1}=p^{n}(p-1)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{mcd}(n,m)=1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\phi(nm)=\phi(n)\phi(m)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Definimos 
+\begin_inset Formula $f:\mathbb{Z}_{nm}^{*}\rightarrow\mathbb{Z}_{n}^{*}\times\mathbb{Z}_{m}^{*}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(x)=(x_{n},x_{m})$
+\end_inset
+
+, donde 
+\begin_inset Formula $x_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{m}$
+\end_inset
+
+ son los restos de dividir 
+\begin_inset Formula $x$
+\end_inset
+
+ entre 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+, respectivamente.
+ Para abreviar asumimos que está bien definida, y pasamos a ver que es biyectiva.
+ Si 
+\begin_inset Formula $f(x)=(x_{n},x_{m})=f(y)$
+\end_inset
+
+ entonces 
+\begin_inset Formula $x\equiv y\,(m)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\equiv y\,(n)$
+\end_inset
+
+, luego 
+\begin_inset Formula $nm|(x-y)$
+\end_inset
+
+ y en 
+\begin_inset Formula $\mathbb{Z}_{nm}^{*}$
+\end_inset
+
+ es inyectiva.
+ Para ver que es suprayectiva, consideramos 
+\begin_inset Formula $(a,b)\in\mathbb{Z}_{n}^{*}\times\mathbb{Z}_{m}^{*}$
+\end_inset
+
+.
+ Al existir una identidad de Bézout 
+\begin_inset Formula $1=rn+sm$
+\end_inset
+
+, podemos hacer 
+\begin_inset Formula $x=brn+asm$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $x\equiv a\,(n)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\equiv b\,(m)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $m=p_{1}^{n_{1}}\cdots p_{s}^{n_{s}}$
+\end_inset
+
+ es la descomposición de 
+\begin_inset Formula $m$
+\end_inset
+
+ en factores primos, entonces
+\begin_inset Formula 
+\[
+\phi(m)=\prod_{i=1}^{s}p_{i}^{n_{i}-1}(p_{i}-1)=m\left(1-\frac{1}{p_{1}}\right)\cdots\left(1-\frac{1}{p_{s}}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Euler:
+\series default
+ Sea 
+\begin_inset Formula $1<m\in\mathbb{Z}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $a$
+\end_inset
+
+ es coprimo con 
+\begin_inset Formula $m$
+\end_inset
+
+ entonces 
+\begin_inset Formula $a^{\phi(m)}\equiv1\,(m)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Esto equivale a que 
+\begin_inset Formula $\overline{a}^{\phi(m)}=\overline{1}\in\mathbb{Z}_{m}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $\mathbb{Z}_{m}^{*}=\{\overline{x_{1}},\dots,\overline{x_{\phi(m)}}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{a}\cdot\mathbb{Z}_{m}^{*}=\{\overline{a}\overline{x}|\overline{x}\in\mathbb{Z}_{m}^{*}\}$
+\end_inset
+
+.
+ Demostramos que 
+\begin_inset Formula $\overline{a}\cdot\mathbb{Z}_{m}^{*}=\mathbb{Z}_{m}^{*}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $x=\overline{ax_{i}}\in\overline{a}\cdot\mathbb{Z}_{m}^{*}\implies x\overline{a}^{-1}\overline{x_{i}}^{-1}=\overline{1}\implies x\in\mathbb{Z}_{m}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\overline{x_{i}}\in\mathbb{Z}_{m}^{*}\implies\overline{x_{i}}=\overline{a}\,\overline{a}^{-1}\overline{x_{i}}\in\overline{a}\cdot\mathbb{Z}_{m}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Entonces 
+\begin_inset Formula $\prod_{i=1}^{\phi(m)}\overline{x}_{i}=\prod_{i=1}^{\phi(m)}\overline{ax_{i}}=\overline{a}^{\phi(m)}\prod_{i=1}^{\phi(m)}\overline{x_{i}}$
+\end_inset
+
+, y dividiendo entre 
+\begin_inset Formula $\prod_{i=1}^{\phi(m)}\overline{x_{i}}$
+\end_inset
+
+, porque es invertible, se obtiene el resultado.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema Pequeño de Fermat:
+\series default
+ Si 
+\begin_inset Formula $a\in\mathbb{Z}$
+\end_inset
+
+ y 
+\begin_inset Formula $p$
+\end_inset
+
+ es un número primo que no divide a 
+\begin_inset Formula $a$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a^{p-1}\equiv1\,(p)$
+\end_inset
+
+, y para todo 
+\begin_inset Formula $x\in\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $x^{p}\equiv x\,(p)$
+\end_inset
+
+.
+ Esto se deriva del teorema de Euler y de que 
+\begin_inset Formula $\phi(p)=p-1$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/cyn/n8.lyx b/cyn/n8.lyx
new file mode 100644
index 0000000..1249714
--- /dev/null
+++ b/cyn/n8.lyx
@@ -0,0 +1,954 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Polinomios con coeficientes en un cuerpo
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+polinomio
+\series default
+ con coeficientes en el cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ es una expresión de la forma
+\begin_inset Formula 
+\[
+a_{0}+a_{1}X+a_{2}X^{2}+\dots+a_{n}X^{n}=\sum_{i=0}^{n}a_{i}X_{i}
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $a_{0},\dots,a_{n}\in K$
+\end_inset
+
+.
+ El símbolo 
+\begin_inset Formula $X$
+\end_inset
+
+ se llama 
+\series bold
+indeterminada
+\series default
+ y llamamos 
+\series bold
+coeficiente
+\series default
+ de grado 
+\begin_inset Formula $i$
+\end_inset
+
+ a 
+\begin_inset Formula $a_{i}$
+\end_inset
+
+, 
+\series bold
+término independiente
+\series default
+ a 
+\begin_inset Formula $a_{0}$
+\end_inset
+
+ y 
+\series bold
+coeficiente principal
+\series default
+ o 
+\series bold
+líder
+\series default
+ a 
+\begin_inset Formula $a_{n}$
+\end_inset
+
+ si es 
+\begin_inset Formula $a_{n}\neq0$
+\end_inset
+
+.
+ Un polinomio es 
+\series bold
+mónico
+\series default
+ si 
+\begin_inset Formula $a_{n}=1$
+\end_inset
+
+.
+ Los polinomios de forma 
+\begin_inset Formula $a_{0}$
+\end_inset
+
+ se llaman 
+\series bold
+constantes
+\series default
+ y los identificamos con los elementos de 
+\begin_inset Formula $K$
+\end_inset
+
+.
+ El conjunto de todos los polinomios con coeficientes en 
+\begin_inset Formula $K$
+\end_inset
+
+ se denota 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, y dos polinomios 
+\begin_inset Formula $P=a_{0}+\dots+a_{n}X^{n},Q=b_{0}+\dots+b_{m}X^{m}\in K[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $n\leq m$
+\end_inset
+
+ son iguales si 
+\begin_inset Formula $a_{i}=b_{i}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,\dots,n\}$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{j}=0$
+\end_inset
+
+ para 
+\begin_inset Formula $j\in\{n+1,\dots,m\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $P+Q=(a_{0}+b_{0})+(a_{1}+b_{1})X+\dots+(a_{n}+b_{n})X^{n}$
+\end_inset
+
+, y 
+\begin_inset Formula $PQ=c_{0}+c_{1}X+\dots+c_{n+m}X^{n+m}$
+\end_inset
+
+ si 
+\begin_inset Formula $c_{k}=\sum_{i+j=k}a_{i}b_{j}=a_{0}b_{k}+a_{1}b_{k-1}+\dots+a_{k}b_{0}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ es un anillo conmutativo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $P$
+\end_inset
+
+ tiene 
+\series bold
+grado
+\series default
+ 
+\begin_inset Formula $n$
+\end_inset
+
+ si 
+\begin_inset Formula $P=\sum_{i=0}^{n}a_{i}X^{i}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n}\neq0$
+\end_inset
+
+, y se denota con 
+\begin_inset Formula $\text{gr}(P)$
+\end_inset
+
+.
+ Por convención, si 
+\begin_inset Formula $P(X)=0$
+\end_inset
+
+, 
+\begin_inset Formula $\text{gr}(P)=-\infty$
+\end_inset
+
+.
+ Si tomamos el convenio de que 
+\begin_inset Formula $-\infty+n=-\infty$
+\end_inset
+
+, 
+\begin_inset Formula $(-\infty)+(-\infty)=-\infty$
+\end_inset
+
+ y 
+\begin_inset Formula $-\infty<n$
+\end_inset
+
+, se tiene que:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(PQ)=\text{gr}(P)+\text{gr}(Q)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(P+Q)\leq\max\{\text{gr}(P),\text{gr}(Q)\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $PQ=0\implies P=0\lor Q=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists P^{-1}:P^{-1}P=1\iff\text{gr}(P)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Se define el 
+\series bold
+valor de 
+\begin_inset Formula $P(x)$
+\end_inset
+
+ en 
+\begin_inset Formula $b$
+\end_inset
+
+
+\series default
+ como 
+\begin_inset Formula $P(b)=a_{0}+a_{1}b+\dots+a_{n}b^{n}\in K$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $P$
+\end_inset
+
+ define una aplicación 
+\begin_inset Formula $P:K\rightarrow K$
+\end_inset
+
+ que llamamos 
+\series bold
+función polinomial asociada a 
+\begin_inset Formula $K$
+\end_inset
+
+
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de la división:
+\series default
+ Para 
+\begin_inset Formula $A,B\in K[X]$
+\end_inset
+
+ existen dos únicos polinomios, 
+\begin_inset Formula $Q$
+\end_inset
+
+ (
+\series bold
+cociente
+\series default
+) y 
+\begin_inset Formula $R$
+\end_inset
+
+ (
+\series bold
+resto
+\series default
+) en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ tales que 
+\begin_inset Formula $A=BQ+R$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(R)<\text{gr}(B)$
+\end_inset
+
+.
+ 
+\series bold
+Teorema del resto:
+\series default
+ El resto de la división de 
+\begin_inset Formula $P/X-a$
+\end_inset
+
+ es 
+\begin_inset Formula $P(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Decimos que 
+\begin_inset Formula $A$
+\end_inset
+
+ divide a 
+\begin_inset Formula $B$
+\end_inset
+
+, o que 
+\begin_inset Formula $B$
+\end_inset
+
+ es múltiplo de 
+\begin_inset Formula $A$
+\end_inset
+
+ (
+\begin_inset Formula $A|B$
+\end_inset
+
+) si 
+\begin_inset Formula $\exists C:B=AC$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A|B\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $A$
+\end_inset
+
+ es un 
+\series bold
+divisor
+\series default
+ de 
+\begin_inset Formula $B$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A|B\land A|C\implies A|PB+QC$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A|B\land B|C\implies A|C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A|B\land B|A\implies\exists\mu\in K\backslash\{0\}:A=\mu B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Los polinomios de la forma 
+\begin_inset Formula $\lambda A$
+\end_inset
+
+ para 
+\begin_inset Formula $0\neq\lambda\in K$
+\end_inset
+
+ se llaman 
+\series bold
+polinomios asociados
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Cada polinomio tiene un único polinomio asociado mónico.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D$
+\end_inset
+
+ es el máximo común divisor de 
+\begin_inset Formula $A,B\in K[X]$
+\end_inset
+
+ si 
+\begin_inset Formula $D|A,B$
+\end_inset
+
+, 
+\begin_inset Formula $S|A,B\implies S|D$
+\end_inset
+
+ y 
+\begin_inset Formula $D$
+\end_inset
+
+ es mónico.
+ Si 
+\begin_inset Formula $D'$
+\end_inset
+
+ verifica las dos primeras condiciones, entonces es asociado a 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ Además, si 
+\begin_inset Formula $A,B\neq0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $D$
+\end_inset
+
+ es el único polinomio mónico de grado mínimo que es combinación lineal
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A,B\in K[X]$
+\end_inset
+
+ son coprimos o primos entre sí si 
+\begin_inset Formula $\text{mcd}(A,B)=1$
+\end_inset
+
+, es decir, si existen 
+\begin_inset Formula $S,T\in K[X]$
+\end_inset
+
+ tales que 
+\begin_inset Formula $SA+TB=1$
+\end_inset
+
+.
+ En tal caso, 
+\begin_inset Formula $A|BC\implies A|C$
+\end_inset
+
+.
+ Además, si 
+\begin_inset Formula $A,B\in K[X]$
+\end_inset
+
+ con alguno de los dos no nulo y 
+\begin_inset Formula $D=\text{mcd}(A,B)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\frac{A}{D}$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{B}{D}$
+\end_inset
+
+ son coprimos.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ es el mínimo común múltiplo de 
+\begin_inset Formula $A,B\in K[X]$
+\end_inset
+
+ si 
+\begin_inset Formula $A,B|M$
+\end_inset
+
+, 
+\begin_inset Formula $A,B|N\implies M|N$
+\end_inset
+
+ y 
+\begin_inset Formula $M$
+\end_inset
+
+ es mónico.
+ Si 
+\begin_inset Formula $M'$
+\end_inset
+
+ cumple las dos primeras condiciones, entonces es asociado a 
+\begin_inset Formula $M$
+\end_inset
+
+.
+ Además, existe 
+\begin_inset Formula $\mu\in K$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\text{mcm}(A,B)=\mu\frac{AB}{\text{mcd}(A,B)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Raíces de polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $r\in K$
+\end_inset
+
+ es una 
+\series bold
+raíz
+\series default
+ de 
+\begin_inset Formula $P\in K[X]$
+\end_inset
+
+ si 
+\begin_inset Formula $P(r)=0$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\frac{p}{q}$
+\end_inset
+
+, con 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+ coprimos, es raíz de 
+\begin_inset Formula $P$
+\end_inset
+
+, entonces 
+\begin_inset Formula $p|a_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $q|a_{n}$
+\end_inset
+
+.
+ La 
+\series bold
+regla de Ruffini
+\series default
+ se basa en que 
+\begin_inset Formula $a$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $P$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $X-a|P$
+\end_inset
+
+.
+ Así, decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ es una raíz de 
+\series bold
+multiplicidad 
+\begin_inset Formula $s\geq1$
+\end_inset
+
+
+\series default
+ de 
+\begin_inset Formula $P$
+\end_inset
+
+ si 
+\begin_inset Formula $(X-a)^{s}|P$
+\end_inset
+
+ pero no 
+\begin_inset Formula $(X-a)^{s+1}|P$
+\end_inset
+
+.
+ Una raíz es 
+\series bold
+múltiple
+\series default
+ si tiene multiplicidad mayor que 1, de lo contrario es una raíz 
+\series bold
+simple
+\series default
+.
+ Si 
+\begin_inset Formula $\text{gr}(P)=n\neq-\infty$
+\end_inset
+
+, entonces 
+\begin_inset Formula $P$
+\end_inset
+
+ tiene a lo sumo 
+\begin_inset Formula $n$
+\end_inset
+
+ raíces en 
+\begin_inset Formula $K$
+\end_inset
+
+, contando cada raíz tantas veces como su multiplicidad.
+ Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(P)=n$
+\end_inset
+
+ y existen 
+\begin_inset Formula $m>n$
+\end_inset
+
+ raíces de 
+\begin_inset Formula $P$
+\end_inset
+
+ en 
+\begin_inset Formula $K$
+\end_inset
+
+, entonces 
+\begin_inset Formula $P=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(P)=n\geq0$
+\end_inset
+
+ y existen 
+\begin_inset Formula $a_{1},\dots,a_{m}\in K$
+\end_inset
+
+ tales que 
+\begin_inset Formula $P(a_{i})=Q(a_{i})$
+\end_inset
+
+ con 
+\begin_inset Formula $m>n$
+\end_inset
+
+, entonces 
+\begin_inset Formula $P=Q$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo infinito y 
+\begin_inset Formula $P,Q\in K[X]$
+\end_inset
+
+ son distintos, entonces las funciones 
+\begin_inset Formula $P,Q:K\rightarrow K$
+\end_inset
+
+ son distintas.
+\end_layout
+
+\begin_layout Enumerate
+Sea 
+\begin_inset Formula $P=a_{0}+\dots+a_{n}X^{n}\in K[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{gr}(P)=n$
+\end_inset
+
+ y raíces 
+\begin_inset Formula $r_{1},\dots,r_{n}$
+\end_inset
+
+ (no necesariamente distintas), entonces 
+\begin_inset Formula $P=a_{n}(X-r_{1})\cdots(X-r_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Factorización y raíces de polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $P\in K[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{gr}(P)>0$
+\end_inset
+
+ es 
+\series bold
+irreducible
+\series default
+ o 
+\series bold
+primo
+\series default
+ si 
+\begin_inset Formula $Q|P\implies\text{gr}(Q)=0\lor\exists k\in K:Q=kP$
+\end_inset
+
+.
+ Así:
+\begin_inset Formula 
+\[
+P\text{ es irreducible}\iff(P|QR\implies P|Q\lor P|R)\iff(P|Q_{1}\cdots Q_{n}\implies\exists i:P|Q_{i})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema:
+\series default
+ Todo 
+\begin_inset Formula $P\in K[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{gr}(P)\geq1$
+\end_inset
+
+ factoriza como producto de polinomios irreducibles, y esta factorización
+ es única salvo asociados y orden.
+\end_layout
+
+\begin_layout Section
+Polinomios irreducibles en 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{C}[X]$
+\end_inset
+
+.
+ Teorema Fundamental del Álgebra
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+Teorema Fundamental del Álgebra
+\series default
+ afirma que todo 
+\begin_inset Formula $P\in\mathbb{C}[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{gr}(P)>0$
+\end_inset
+
+ tiene al menos una raíz en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+.
+ Así:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P\in\mathbb{C}[X]$
+\end_inset
+
+ es irreducible si y sólo si 
+\begin_inset Formula $\text{gr}(P)=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall P\in\mathbb{C}[X],\text{gr}(P)=n\geq1,\exists r,r_{1},\dots,r_{n}\in\mathbb{C}:P=r(X-r_{1})\cdots(X-r_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $P\in\mathbb{R}[X]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overline{z}$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P\in\mathbb{R}[X]$
+\end_inset
+
+ es irreducible, entonces, o 
+\begin_inset Formula $\text{gr}(P)=1$
+\end_inset
+
+, o 
+\begin_inset Formula $\text{gr}(P)=2$
+\end_inset
+
+ y no tiene raíces reales.
+\end_layout
+
+\end_body
+\end_document
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