Bijection iff Left and Right Inverse/Corollary

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Corollary to Bijection iff Left and Right Inverse

Let $f: S \to T$ and $g: T \to S$ be mappings such that:

$g \circ f = I_S$
$f \circ g = I_T$

Then both $f$ and $g$ are bijections.


Proof

Suppose we have such mappings $f$ and $g$ with the given properties.

From Bijection iff Left and Right Inverse, we have that $f$ is a bijection, by considering $g = g_1$ and $g = g_2$.

It directly follows that by setting $g = f, f = g_1, f = g_2$, the result Bijection iff Left and Right Inverse can be used the other way about.

$\blacksquare$


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