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

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