Question

* is defined on the set of real numbers by $a*b = 1 + ab$. Then the operation * is

• A. Commutative but not associative
• B. Associative but not commutative
• C. Neither commutative nor associative
• D. Both commutative and associative

Answer 1

Answer: (A)

Commutative but not associative

Answer 2

We have $a*b = 1 + ab = 1 + ba = b*a$

So, * is commutative on R.

For any, a, b, c $\in$ R, we have

$(a*b)*c = (1 + ab)*c = 1 + (1 + ba)c = 1 + c + abc$and $a*(b*c) = a*(1 + bc) = 1 + a(1 + bc) = 1 + a + abc\therefore$ $(a*b)*c \neq a*(b*c)$

So, * is not associative on R