Bijection

A function $f:X\to Y$ is a bijection if it is both injective (one-to-one) and surjective (onto):

• Injective: $f(x_1)=f(x_2)\Rightarrow x_1=x_2$ - distinct inputs go to distinct outputs.
• Surjective: for every $y\in Y$ there exists $x\in X$ with $f(x)=y$ - every $y$ is hit.

Equivalently, $f$ is a bijection if it has a two-sided inverse $f^{-1}:Y\to X$ satisfying $f\circ f^{-1}={\mathrm{id}}_Y$ and $f^{-1}\circ f={\mathrm{id}}_X$.

For finite sets, $f:X\to Y$ is a bijection if and only if $|X|=|Y|$ and $f$ is either injective or surjective (the other condition follows automatically).

In Group Theory

Bijections appear all over the course:

• A Permutation is a bijection from a set to itself.
• An Isomorphism is a bijective Homomorphism.
• The map $g\mapsto \lambda (g)$ from a Group action is realised as a homomorphism into the symmetric group $\mathrm{Sym}(X)$ - itself a group of bijections.