Definition:Bijection

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Definition

Definition 1

A mapping $f: S \to T$ is a bijection if and only if:

$f$ is both a surjection and an injection.


Definition 2

A mapping $f: S \to T$ is a bijection if and only if:

$f$ has both a left inverse and a right inverse.


Definition 3

A mapping $f: S \to T$ is a bijection if and only if:

the inverse $f^{-1}$ of $f$ is a mapping from $T$ to $S$.


Definition 4

A mapping $f \subseteq S \times T$ is a bijection if and only if:

for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.


Definition 5

A relation $f \subseteq S \times T$ is a bijection if and only if:

$(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\left({x, y}\right) \in f$
$(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.


Also known as

The terms

biunique correspondence
bijective correspondence

are sometimes seen for bijection.

Authors who prefer to limit the jargon of mathematics tend to use the term one-one and onto mapping for bijection.

If a bijection exists between two sets $S$ and $T$, then $S$ and $T$ are said to be in one-to-one correspondence.

Occasionally you will see the term set isomorphism, but the term isomorphism is usually reserved for mathematical structures of greater complexity than a set.

Some authors, developing the concept of inverse mapping independently of that of the bijection, call such a mapping invertible.


The symbol $f: S \leftrightarrow T$ is sometimes seen to denote that $f$ is a bijection from $S$ to $T$.

Also seen sometimes is the notation $f: S \cong T$ or $S \stackrel f \cong T$ but this is cumbersome and the symbol has already got several uses.


Also see


  • Results about bijections can be found here.


Basic Properties of a Bijection



Sources