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Commutative operation

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

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

Further examples of commutative binary operations include addition and multiplication of real and 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.

See also: Associativity, Distributive property