Factoring a function through the projection of an equivalence relation induced by that function yields an injection

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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 [ilmath]f:X\rightarrow Y[/ilmath] through [ilmath]\pi:X\rightarrow \frac{X}{\sim} [/ilmath] to yield an injective map:

• [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.

Proof

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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