Bijection

$ \def\Id {\mathop{\rm Id}} $

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 $.

Equivalent condition

A mapping is bijective if and only if

• it has left-sided and right-sided inverses

and therefore if and only if

• there is a unique (two-sided) inverse mapping $ f^{-1} $ such that $ f^{-1} \circ f = \Id_A $ and $ f \circ f^{-1} = \Id_B $.

Application

Bijections are essential for the theory of cardinal numbers:
Two sets have the same number of elements (the same cardinality), if there is a bijective mapping between them.
By the Schröder-Bernstein theorem — and not depending on the Axiom of Choice — a bijective mapping between two sets $A$ and $B$ exists if there are injective mappings both from $A$ to $B$ and from $B$ to $A$.

Related notions

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, and—if domain and range coincide—automorphism.