A mapping $f: S \to T$ is a bijection if and only if:
That is, if and only if $f$ admits an inverse.
Also known as
- 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.
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 .
- Inverse of Bijection is Bijection, where it is shown that this inverse mapping is also a bijection.
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.11$: Relations: Theorem $11.11$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.11$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics: bijection
- Barile, Margherita. "Bijective." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Bijective.html