# Definition:Bijection/Class Theory

## Definition

Let $A$ and $B$ be classes.

Let $f: A \to B$ be a class mapping from $A$ to $B$.

Then $f$ is said to be a **bijection** if and only if:

- $f$ is both a class injection and a class surjection.

## 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`

.

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

- Results about
**bijections**can be found**here**.