# 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.