Definition:Bijection/Definition 5

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Definition

A relation $f \subseteq S \times T$ is a bijection if and only if:

$(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\tuple {x, y} \in f$
$(2): \quad$ 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

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 has already got several uses.


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