Commutativity
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. Contrariwise if this property is not established the operation is said to be noncommutative. 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 (ab), functional composition (f(g(x)), and tetration (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 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, commutant, commutator.
