Difference between revisions of "Factoring a function through the projection of an equivalence relation induced by that function yields an injection"
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{{float-right|{{/Diagram}}}}Let {{M|X}} and {{M|Y}} be [[sets]], let {{M|f:X\rightarrow Y}} be any [[function]] between them, and let {{M|\sim\subseteq X\times X}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by a function|induced by the function {{M|f}}]], recall that means: | {{float-right|{{/Diagram}}}}Let {{M|X}} and {{M|Y}} be [[sets]], let {{M|f:X\rightarrow Y}} be any [[function]] between them, and let {{M|\sim\subseteq X\times X}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by a function|induced by the function {{M|f}}]], recall that means: | ||
* {{M|1=\forall x,x'\in X[x\sim x'\iff f(x)=f(x')]}} | * {{M|1=\forall x,x'\in X[x\sim x'\iff f(x)=f(x')]}} | ||
− | Then we claim we can {{link|factor|function}}<ref group="Note">{{AKA}}: {{link|passing to the quotient|function}}</ref> {{M|f:X\rightarrow Y}} through {{M|\pi:X\rightarrow \frac{X}{\sim} }}<ref group="Note">the [[canonical projection of the equivalence relation]], given by {{M|\pi:x\mapsto [x]}} where {{M|[x]}} denotes the [[equivalence class]] containing {{M|x}}</ref> to {{underline|yield | + | Then we claim we can {{link|factor|function}}<ref group="Note">{{AKA}}: {{link|passing to the quotient|function}}</ref> {{M|f:X\rightarrow Y}} through {{M|\pi:X\rightarrow \frac{X}{\sim} }}<ref group="Note">the [[canonical projection of the equivalence relation]], given by {{M|\pi:x\mapsto [x]}} where {{M|[x]}} denotes the [[equivalence class]] containing {{M|x}}</ref> to {{underline|yield a unique [[injective]]}} map<ref>[[File:MondTop2016ex1.pdf]]</ref>: |
* {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} | * {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} | ||
Furthermore, if {{M|f:X\rightarrow Y}} is [[surjective]] then {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} is not only [[injective]] but [[surjective]] to, that is: {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} is a [[bijection]]<ref group="Note">See "''[[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]]''" for details</ref>. | Furthermore, if {{M|f:X\rightarrow Y}} is [[surjective]] then {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} is not only [[injective]] but [[surjective]] to, that is: {{M|\tilde{f}:\frac{X}{\sim}\rightarrow Y}} is a [[bijection]]<ref group="Note">See "''[[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]]''" for details</ref>. | ||
<div style="clear:both;"></div> | <div style="clear:both;"></div> | ||
− | |||
==Proof== | ==Proof== | ||
{{Requires proof|grade=A*|msg=Do this now, just saving work}} | {{Requires proof|grade=A*|msg=Do this now, just saving work}} | ||
+ | * Note to self - uniqueness comes from that we're [[factor (function)|factoring]] through a [[surjective]] map (namely, {{M|\pi}}), we only really have to show the result is injective. | ||
==See also== | ==See also== | ||
* [[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]] | * [[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]] |
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Contents
Statement
Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] be sets, let [ilmath]f:X\rightarrow Y[/ilmath] be any function between them, and let [ilmath]\sim\subseteq X\times X[/ilmath] denote the equivalence relation induced by the function [ilmath]f[/ilmath], recall that means:- [ilmath]\forall x,x'\in X[x\sim x'\iff f(x)=f(x')][/ilmath]
Then we claim we can factor[Note 1] [ilmath]f:X\rightarrow Y[/ilmath] through [ilmath]\pi:X\rightarrow \frac{X}{\sim} [/ilmath][Note 2] to yield a unique injective map[1]:
- [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath]
Furthermore, if [ilmath]f:X\rightarrow Y[/ilmath] is surjective then [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is not only injective but surjective to, that is: [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is a bijection[Note 3].
Proof
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- Note to self - uniqueness comes from that we're factoring through a surjective map (namely, [ilmath]\pi[/ilmath]), we only really have to show the result is injective.
See also
Notes
- ↑ AKA: passing to the quotient
- ↑ the canonical projection of the equivalence relation, given by [ilmath]\pi:x\mapsto [x][/ilmath] where [ilmath][x][/ilmath] denotes the equivalence class containing [ilmath]x[/ilmath]
- ↑ See "If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection" for details
References
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