# 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

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