# 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 set $A$ onto itself is called a *permutation* of $A$.

A bijective homomorphism is called *isomorphism*,
and—if domain and range coincide—*automorphism*.

**How to Cite This Entry:**

Bijection.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Bijection&oldid=30987