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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 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