The Commutative operation reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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. If x * y = y * x for a particular choice of elements x and y, then x and y are said to commute.

The most well known examples of commutative binary operations are addition (a+b) and multiplication (a*b) 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 the binary operations that are not commutative are subtraction (a-b), division (a/b), exponentiation (a^n), functional composition (f(g(x)), and super-exponentiation (a@b).

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 functionss.

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, commutant, commutator.

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