Commit inicial, primer cuatrimestre.

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+#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