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')]}}
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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 an [[injective]]}} map:
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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>
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==Proof==
 
==Proof==
 
{{Requires proof|grade=A*|msg=Do this now, just saving work}}
 
{{Requires proof|grade=A*|msg=Do this now, just saving work}}
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* 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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Statement

Commutative diagram showing the situation
Let X and Y be sets, let f:X→Y be any function between them, and let ∼⊆X×X denote the equivalence relation induced by the function f, recall that means:
  • ∀x,x′∈X[x∼x′⟺f(x)=f(x′)]

Then we claim we can factor[Note 1] f:X→Y through π:X→X∼[Note 2] to yield a unique injective map[1]:

  • ˜f:X∼→Y

Furthermore, if f:X→Y is surjective then ˜f:X∼→Y is not only injective but surjective to, that is: ˜f:X∼→Y 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, π), we only really have to show the result is injective.

See also

Notes

  1. Jump up ↑ AKA: passing to the quotient
  2. Jump up ↑ the canonical projection of the equivalence relation, given by π:x↦[x] where [x] denotes the equivalence class containing x
  3. Jump up ↑ 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

  1. Jump up ↑ File:MondTop2016ex1.pdf


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