# Definition:Bijection/Definition 3

## Definition

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

That is, if and only if $f$ admits an inverse.

## 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

- Inverse of Bijection is Bijection, where it is shown that this inverse mapping is also a
**bijection**.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.11$: Relations: Theorem $11.11$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 7.11$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*:**bijection**

- Barile, Margherita. "Bijective." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Bijective.html