Question

Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.

Options

• Commutative but not associative

• Associative but not commutative

• Neither commutative nor associative

• Both commutative and associative

Answer 1

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

Explanation:

Given that * is a binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R

So, we have a * b = ab + 1 = b * a

So, * is a commutative binary operation.

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

Also,

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

Thus, a * (b * c) ≠ (a * b) * c

Hence, * is not associative.

Therefore, * is commutative but not associative.