Definition:Bijection/Definition 5
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 $\cong$ 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
Sources
- 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Equivalent Sets: Definition: $1.4$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings