View source for Equivalence relation induced by a function

{{Stub page|grade=B|msg=Something weird happened with [[every surjective map gives rise to an equivalence relation]] this page is what it SHOULD be. I also have a reference, granted not that strong of one}}
{{Requires references|grade=A|msg=Homework assignment isn't sufficient}}
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==Statement==
Let {{M|X}} and {{M|Y}} be [[sets]] and let {{M|f:X\rightarrow Y}} be any [[mapping]] between them. Then {{M|f}} induces an [[equivalence relation]], {{M|\sim\subseteq X\times X}} where<ref>[[File:MondTop2016ex1.pdf]]</ref>:
* for {{M|x_1,x_2\in X}} we say {{M|x_1\sim x_2}} if {{M|1=f(x_1)=f(x_2)}}
Note that {{M|f}} may be [[factor (function)|factored]] through the [[canonical projection of an equivalence relation]] to yield an injection. Furthermore if {{M|f}} is [[surjective]], then so is the induced map, and then the induced map is a bijection.
* See: [[factoring a function through the projection of an equivalence relation induced by that function yields an injection]] and [[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]]

==Proof==
{{Requires proof|grade=D|easy=true|msg=Easy proof, marked as such. Just gotta show it's an [[equivalence relation]]}}
==See also==
* [[Factoring a function through the projection of an equivalence relation induced by that function yields an injection]]
** [[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]]
{{Todo|Link to continuous version ([[File:MondTop2016ex1.pdf]] - Q5)}}

==References==
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{{Definition|Elementary Set Theory|Set Theory}}