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.

**Solution:**

**(1) **“” is commutative in because if , then

**(2)** “” is associative in because if then