Difference between revisions of "Factoring a function through the projection of an equivalence relation induced by that function yields an injection"

From Maths
Jump to: navigation, search
m (→‎Statement: Fix weird sentence)
m (Reference, proof note)
Line 4: Line 4:
 
{{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 an [[injective]]}} map:
+
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]]

Revision as of 12:59, 9 October 2016

Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Flesh out and demote

Statement

[ilmath]\xymatrix{ X \ar[r]^f \ar[d]_{\pi} & Y \\ \frac{X}{\sim} \ar@{.>}[ur]_{\tilde{f} } }[/ilmath]
Commutative diagram showing the situation
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

Grade: A*
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Do this now, just saving work
  • 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

  1. ↑ AKA: passing to the quotient
  2. ↑ 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]
  3. ↑ 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. ↑ File:MondTop2016ex1.pdf


Retrieved from "https://www.maths.kisogo.com/index.php?title=Factoring_a_function_through_the_projection_of_an_equivalence_relation_induced_by_that_function_yields_an_injection&oldid=3249"