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