Equivalence relation induced by a function

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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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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:

• 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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Easy proof, marked as such. Just gotta show it's an equivalence relation

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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

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TODO: Link to continuous version (File:MondTop2016ex1.pdf - Q5)

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