Definition:Bijection

Definition
A mapping $f: S \to T$ is a bijection iff $f$ is both a surjection and an injection.

It is clear that a bijection is a relation which is:
 * left-total
 * right-total
 * functional (many-to-one)
 * injective (one-to-many).

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.

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

 * Definition:Injection
 * Definition:Surjection
 * Definition:Permutation/Mapping


 * Definition:Set Equivalence

Basic Properties of a Bijection

 * In Bijection iff Left and Right Inverse, it is shown that a mapping $f$ is a bijection iff 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 iff it is both left cancellable and a right cancellable.