Associativity

In mathematics, associative binary operation on a set S is a binary operation * satisfying the law (x * y) * z = x * (y * z) for all members x, y, and z of S.

The most commonly known examples of associativity are addition and multiplication of real numbers; for example:

(7 + 3) + 9 = 7 + (3 + 9), since the expression on the left evaluates to 10 + 9 = 19, which the expression on the right evaluates to 7 + 12 = 19, the same value;
(10 × 5) × 3 = 10 × (5 × 3), since the expression on the left evaluates to 50 × 3 = 150, while the expression on the right evalutes to 10 × 15 = 150.

Among widely known binary operations that are not associative are subtraction (i.e., x − (y − z) is not the same as (x − y) − z), division, exponentiation, and the operation of taking the average of two numbers. The last is an example of a binary operation that is commutative but not associative.

Other examples of associative binary operations include addition and multiplication of complex numbers and square matrices; addition of vectors; and intersection and union of sets. Also, if M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative.

A set with an associative binary operation on it is called a semigroup; monoids and groupss are examples of semigroups.