Let * be binary operation defined on R by a * b = 1 + ab, ∀a, b ∈R. Then the operation * is
(i) commutative but not associative

(ii) associative but not commutative

(iii) neither commutative nor associative

(iv) both commutative and associative

Question

Philoid · Accepted Answer

Given that,

‘*’ be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R

A binary operation ‘*’ is commutative if a*b = b*a ∀ a, b ∈ R

Now,

a*b = 1+ab = 1+ba [∵ ab=ba since multiplication is commutative

on R]

⇒ 1+ba = b*a

∴ a*b = b*a ∀ a, b ∈ R

So, ‘*’ is commutative on R.

A binary operation ‘*’ is associative if (a*b)*c = a*(b*c) ∀ a, b,c ∈ R

Now,

(a*b)*c = (1+ab)*c = 1+(1+ab)c = 1+c+abc

a*(b*c) = a*(1+bc) = 1+a(1+bc) = 1+a+abc

∴ (a*b)*c ≠ a*(b*c)

So, ‘*’ is not associative on R.

Hence, ‘*’ is commutative but not associative on R.

Let * be binary operation defined on R by a * b = 1 + ab, ∀a, b ∈R. Then the operation * is(i) commutative but not associative(ii) associative but not commutative(iii) neither commutative nor associative(iv) both commutative and associative