, especially abstract algebra
, a binary operation
* on a set S
if, for all x
The most commonly known examples of commutativity are addition and multiplication of natural numbers; for example:
- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6)
Further examples of commutative binary operations include addition and multiplication of real
and complex numbers
, addition of vectors
, and intersection
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