Question

Let * be a binary operation defined on the set of rational numbers defined by a*b=ab+7 verify wheather * is a binary operation

Answer 1

To check: Whether * is commutative and associative.

Solution:

Step 1. Commutative checking

Here, a * b = ab + 7

= ba + 7

= b * a

Since a * b = b * a, * is commutative.

Step 2. Associative checking

Here, (a * b) * c = (ab + 7) * c

= (ab + 7) c + 7

= abc + 7c + 7

and a * (b * c) = a * (bc + 7)

= a (bc + 7) + 7

= abc + 7a + 7

Since (a * b) * c ≠ a * (b * c), * is not associative.

Remark: If * was defined by a * b = (ab)/7,

then (a * b) *c = (ab) / 7 * c

= {(ab) / 7 × c} / 7

= (abc) / 49

and a * (b * c) = a * (bc)/7

= {a (bc)/7} / 7

= (abc) / 49

Since (a * b) * c = a * (b * c), * is a associative.