Bijection

A function (or mapping) is called bijective if it is both one-to-one and onto, i.e., if it is both injective and surjective.

In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is

a bijective function or a bijection

if and only if

$ f(A) = B $ and $ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $ for all $ a_1, a_2 \in A $.

In certain contexts, a bijective mapping of a (countable) set $A$ onto itself is called a permutation of $A$.

A bijective homomorphism is called isomorphism.