# Definition:Bijection/Definition 4

## Definition

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 $\tuple {x, y} \in f$.

## Also known as

The terms

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`

.

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

.

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

.

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 2$: Product sets, mappings - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 13$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 22$: Injections; bijections; inverse of a bijection - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*:**bijection** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.6$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?