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:
- 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 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