Equivalence relation induced by a function
From Maths
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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
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Homework assignment isn't sufficient
Contents
Statement
Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] be sets and let [ilmath]f:X\rightarrow Y[/ilmath] be any mapping between them. Then [ilmath]f[/ilmath] induces an equivalence relation, [ilmath]\sim\subseteq X\times X[/ilmath] where^{[1]}:
 for [ilmath]x_1,x_2\in X[/ilmath] we say [ilmath]x_1\sim x_2[/ilmath] if [ilmath]f(x_1)=f(x_2)[/ilmath]
Note that [ilmath]f[/ilmath] may be factored through the canonical projection of an equivalence relation to yield an injection. Furthermore if [ilmath]f[/ilmath] 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
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This proof has been marked as an page requiring an easy proof
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Easy proof, marked as such. Just gotta show it's an equivalence relation
This proof has been marked as an page requiring an easy proof
See also
TODO: Link to continuous version (File:MondTop2016ex1.pdf  Q5)
References
