Bijection

Last updated Nov 1, 2022

A Function that is both an Injection and a Surjection.

1. Every Bijection $f: X \to Y$ has a Function Inverse. This is because for each $y \in Y$, there exists $x \in X$ so that $f(x) = y$, since $f$ is a Surjection. This $x$ is unique because $f$ is an Injection, so we can define $f^{-1}(y) := x$. Then $f \circ f^{-1}(y) = y$ and $f^{-1} \circ f(x) = x$ for all $y \in Y$, $x \in X$.