Bijective function

A bijective function is a function $f:A\to B$ that is both injective and surjective .

A bijection sets up a perfect pairing between elements of the domain and codomain, and it is exactly the situation in which an inverse function exists. Two sets have the same cardinality precisely when there is a bijection between them.

Examples:

• The function $f:\mathbb{Z}\to\mathbb{Z}$ given by $f(n)=n+1$ is bijective.
• If $A=\{1,2,3\}$ and $B=\{a,b,c\}$, the function defined by $1\mapsto a$, $2\mapsto b$, $3\mapsto c$ is bijective.