Editing Bijection (section)

==Definition==
For a [[binary relation]] pairing elements of set ''X'' with elements of set ''Y'' to be a bijection, four properties must hold:
# each element of ''X'' must be paired with at least one element of ''Y'',
# no element of ''X'' may be paired with more than one element of ''Y'',
# each element of ''Y'' must be paired with at least one element of ''X'', and
# no element of ''Y'' may be paired with more than one element of ''X''.

Satisfying properties (1) and (2) means that a pairing is a [[Function (mathematics)|function]] with [[Domain of a function|domain]] ''X''. It is more common to see properties (1) and (2) written as a single statement: Every element of ''X'' is paired with exactly one element of ''Y''. Functions which satisfy property (3) are said to be "[[onto]] ''Y'' " and are called [[Surjective function|surjections]] (or ''surjective functions''). Functions which satisfy property (4) are said to be "[[one-to-one function]]s" and are called [[Injective function|injections]] (or ''injective functions'').<ref>There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a ''total relation'' and a relation satisfying (2) is a ''single valued relation''.</ref> With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".<ref>{{Cite web|url=https://brilliant.org/wiki/bijection-injection-and-surjection/|title=Bijection, Injection, And Surjection {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-07}}</ref>