Definition:Bijection/Also known as

Definition

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 .

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Definition $3$