# Definition:Bijection/Definition 5

## Definition

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

## 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 from 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 $\cong$ already has several uses.

In the context of class theory, a **bijection** is often seen referred to as a **class bijection**.

## Technical Note

The $\LaTeX$ code for \(f: S \leftrightarrow T\) is `f: S \leftrightarrow T`

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The $\LaTeX$ code for \(f: S \cong T\) is `f: S \cong T`

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The $\LaTeX$ code for \(S \stackrel f \cong T\) is `S \stackrel f \cong T`

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## Also see

## Sources

- 1971: Patrick J. Murphy and Albert F. Kempf:
*The New Mathematics Made Simple*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Equivalent Sets: Definition: $1.4$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings