Question

Let ‘&’ be a binary operation defined on the set N. Which of the following definitions is commutative but not associative?

(a) a & b=a-b

(b) a & b=a+b

(c) a & b=ab – 8

(d) a & b=ab

I have been asked this question in a job interview.

This interesting question is from Binary Operations in portion Relations and Functions of Mathematics – Class 12

Answer 1

Right option is (c) a & b=ab – 8

The explanation: The binary operation ‘&’ is commutative but not associative for a*b=ab-8.

For Commutative: a & b=ab-8 and b & a=ba-8

ab-8=ba-8. Hence, a & b=ab-8 is commutative.

For Associative: (a &b)& c=(ab-8)& c=(ab-8)c-8=abc-8c-8=abc-8c-8.

a& (b &c)=a&(bc-8)=a(bc-8)-8=abc-8a-8.

⇒(a&b) & c≠a& (b& c). Hence, the function is not associative.