Equivalence relation and a function
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Suppose $A$ is a nonempty set and $R$ is an equivalence relation on $A$ . Show that there is a function $f$ with $A$ as its domain such that $(x,y) in R$ if and only if $f(x)=f(y)$
discrete-mathematics relations
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
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add a comment |
$begingroup$
Suppose $A$ is a nonempty set and $R$ is an equivalence relation on $A$ . Show that there is a function $f$ with $A$ as its domain such that $(x,y) in R$ if and only if $f(x)=f(y)$
discrete-mathematics relations
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
$endgroup$
$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58
add a comment |
$begingroup$
Suppose $A$ is a nonempty set and $R$ is an equivalence relation on $A$ . Show that there is a function $f$ with $A$ as its domain such that $(x,y) in R$ if and only if $f(x)=f(y)$
discrete-mathematics relations
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
$endgroup$
Suppose $A$ is a nonempty set and $R$ is an equivalence relation on $A$ . Show that there is a function $f$ with $A$ as its domain such that $(x,y) in R$ if and only if $f(x)=f(y)$
discrete-mathematics relations
discrete-mathematics relations
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
asked Dec 30 '18 at 9:43
Arben_AjrediniArben_Ajredini
205
205
$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58
add a comment |
$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58
$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58
$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58
add a comment |
1 Answer
1
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Let $equiv$ be an equivalence relation on the set $A$. Let $bar a = {bin Amid aequiv b}$ be the equivalence class of $ain A$. Take the quotient set $bar A = {bar amid ain A}$. Consider the mapping $f:Arightarrowbar A:amapsto bar a$ sending each element of $A$ to its equivalence class. Then for all $a,bin A$, $aequiv b$ iff $bar a = bar b$ iff $f(a)=f(b)$.
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
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1 Answer
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1 Answer
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active
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$begingroup$
Let $equiv$ be an equivalence relation on the set $A$. Let $bar a = {bin Amid aequiv b}$ be the equivalence class of $ain A$. Take the quotient set $bar A = {bar amid ain A}$. Consider the mapping $f:Arightarrowbar A:amapsto bar a$ sending each element of $A$ to its equivalence class. Then for all $a,bin A$, $aequiv b$ iff $bar a = bar b$ iff $f(a)=f(b)$.
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
$endgroup$
add a comment |
$begingroup$
Let $equiv$ be an equivalence relation on the set $A$. Let $bar a = {bin Amid aequiv b}$ be the equivalence class of $ain A$. Take the quotient set $bar A = {bar amid ain A}$. Consider the mapping $f:Arightarrowbar A:amapsto bar a$ sending each element of $A$ to its equivalence class. Then for all $a,bin A$, $aequiv b$ iff $bar a = bar b$ iff $f(a)=f(b)$.
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
$endgroup$
add a comment |
$begingroup$
Let $equiv$ be an equivalence relation on the set $A$. Let $bar a = {bin Amid aequiv b}$ be the equivalence class of $ain A$. Take the quotient set $bar A = {bar amid ain A}$. Consider the mapping $f:Arightarrowbar A:amapsto bar a$ sending each element of $A$ to its equivalence class. Then for all $a,bin A$, $aequiv b$ iff $bar a = bar b$ iff $f(a)=f(b)$.
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
$endgroup$
Let $equiv$ be an equivalence relation on the set $A$. Let $bar a = {bin Amid aequiv b}$ be the equivalence class of $ain A$. Take the quotient set $bar A = {bar amid ain A}$. Consider the mapping $f:Arightarrowbar A:amapsto bar a$ sending each element of $A$ to its equivalence class. Then for all $a,bin A$, $aequiv b$ iff $bar a = bar b$ iff $f(a)=f(b)$.
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
answered Dec 30 '18 at 9:52
WuestenfuxWuestenfux
5,2931513
5,2931513
add a comment |
add a comment |
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$begingroup$
Exercise. State and prove the converse.
$endgroup$
– William Elliot
Dec 30 '18 at 9:58