# Algebraic Structure

A non-empty set $$G$$ together with at least one binary operation defined on it is called an algebraic structure. Thus if $$G$$ is a non-empty set and “$$ * $$” is a binary operation on $$G$$, then $$\left( {G, * } \right)$$ is an algebraic structure.

$$\left( {n, + } \right)$$, $$\left( {\mathbb{Z}, + } \right)$$, $$\left( {\mathbb{Z}, – } \right)$$, $$\left( {\mathbb{R}, + , \times } \right)$$

are all algebraic structures. Since addition and multiplication are both binary operations on the set $$\mathbb{R}$$ of real numbers, $$\left( {\mathbb{R}, + , \times } \right)$$ is an algebraic structure equipped with two operations.

**Example:** If the binary operation $$ * $$ on $$\mathbb{Q}$$ the set of rational numbers is defined by

$$a * b = a + b – ab,\forall a,b \in \mathbb{Q}$$

show that $$ * $$ is commutative and associative.

**Solution:**

**(1) **“$$ * $$” is commutative in $$\mathbb{Q}$$ because if $$a,b \in \mathbb{Q}$$, then

$$a * b = a + b – ab = b + a – ba = b * a$$

**(2)** “$$ * $$” is associative in $$\mathbb{Q}$$ because if $$a,b,c \in \mathbb{Q}$$ then

$$a * \left( {b * c} \right) = a * \left( {b + c – bc} \right)$$

$$ = a + \left( {b + c – bc} \right) – a\left( {b + c – bc} \right)$$

$$ = a + b – ab + c – \left( {a + b – ab} \right)c$$

$$ = \left( {a * b} \right) * c$$