A non-empty set together with at least one binary operation defined on it is called an algebraic structure. Thus if is a non-empty set and “” is a binary operation on , then is an algebraic structure.
, , ,
are all algebraic structures. Since addition and multiplication are both binary operations on the set of real numbers, is an algebraic structure equipped with two operations.
Example: If the binary operation on the set of rational numbers is defined by
show that is commutative and associative.
(1) “” is commutative in because if , then
(2) “” is associative in because if then