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.