Commutative operation

In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S.

The most commonly known examples of commutativity are addition and multiplication of real numbers; for example:

4 + 5 = 5 + 4 (since both expressions evaluate to 9)
2 × 3 = 3 × 2 (since both expressions evaluate to 6)

Among commonly known binary operations that are not commutative are subtraction, division, exponentiation, and functional composition.

Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. Important non-commutative operations are the multiplication of matrices and the composition of functions.

An abelian group is a group whose operation is commutative.

A ring is called commutative if its multiplication is commutative, since the addition is commutative in any ring.