In
mathematics, a
binary operation * on a
set S is called
associative if
for all x,
y and
z in
S, (
x *
y) *
z =
x * (
y *
z).
The most commonly known examples of associativity are addition and multiplication of natural 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.
Other examples of associative binary operations include addition and multiplication of real numbers, 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 groups are examples of semigroups.
See also Commutativity, Distributive property, Identity element
All Wikipedia text
is available under the
terms of the GNU Free Documentation License
