commutative

Commutative describes a property of certain binary operations in mathematics where changing the order of the operands does not affect the result. Formally, a binary operation ∘ on a set S is commutative if a ∘ b = b ∘ a for all elements a and b in S.

Common examples include arithmetic on real numbers: addition and multiplication satisfy a + b = b + a and a · b = b · a for any numbers a and b. Other familiar commutative operations include union and intersection of sets. Some logical operations, such as exclusive or (XOR) on bits, are also commutative. By contrast, subtraction and division are not generally commutative, since a − b ≠ b − a and a ÷ b ≠ b ÷ a in general. Matrix multiplication is another well-known noncommutative operation, since AB and BA are not necessarily equal.

In abstract algebra, commutativity is a central structural property. An abelian group is a group whose operation is commutative. A commutative ring is a ring in which both addition and multiplication are commutative. A field is a commutative ring with multiplicative inverses for all nonzero elements. The term “commutative” derives from Latin commutativus, meaning to interchange or exchange.

In broader contexts, commutativity can be discussed in relation to functions, operators, and higher algebra, where diagrams are said to commute if different paths yield the same result.