Definition:Bijection
Contents
Definition
Definition 1
A mapping $f: S \to T$ is a bijection if and only if:
- $f$ is both a surjection and an injection.
Definition 2
A mapping $f: S \to T$ is a bijection if and only if:
- $f$ has both a left inverse and a right inverse.
Definition 3
A mapping $f: S \to T$ is a bijection if and only if:
Definition 4
A mapping $f \subseteq S \times T$ is a bijection if and only if:
- for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.
Definition 5
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 $\left({x, y}\right) \in f$
- $(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.
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 of 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.
Also see
- Results about bijections can be found here.
Basic Properties of a Bijection
- In Bijection iff Left and Right Inverse, it is shown that a mapping $f$ is a bijection if and only if it has both a left inverse and a right inverse, and that these are the same, called the (two-sided) inverse.
- In Bijection iff Inverse is Bijection, it is shown that the inverse mapping $f^{-1}$ of a bijection $f$ is also a bijection, and that it is the same mapping as the (two-sided) inverse.
- In Bijection Composite with Inverse, it is established that the inverse mapping $f^{-1}$ and the (two-sided) inverse are the same thing.
- In Bijection iff Left and Right Cancellable, it is shown that a mapping $f$ is a bijection if and only if it is both left cancellable and right cancellable.
Sources
- Barile, Margherita. "One-to-One." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/One-to-One.html