# 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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## Contents

Note: there is a precursor theorem: Factoring a function through the projection of an equivalence relation induced by that function yields an injection

## Statement

 Commutative diagram showing the situation [ilmath]\xymatrix{ X \ar[r]^f \ar[d]_{\pi} & Y \\ \frac{X}{\sim} \ar@{.>}[ur]_{\tilde{f} } }[/ilmath]
Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] be sets, let [ilmath]f:X\rightarrow Y[/ilmath] be any surjective 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 bijective map[1]:

• [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath]

## Proof

• we get a unique[Note 3] injection [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath]

We also know that:

Thus we see [ilmath]\tilde{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is a unique bijection (it is surjective and injective)

## Notes

1. 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]
2. As [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] is surjective - see factor (function)