Question

Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

Answer 1

On N, the operation * is defined as a * b = a³ + b³.

For, a, b, ∈ N, we have:

a * b = a³ + b³ = b³ + a³ = b * a [Addition is commutative in N]

Therefore, the operation * is commutative.

It can be observed that:

`(1 * 2)* 3 = (1^3 + 2^3)*3 = 9 * 3 =9^3 + 3^3 = 729 + 27 = 756`

`1 * (2 * 3) = 1 * (2^3 + 3^3) = 1 * (8 + 27) = 1^3 + 35^3 = 1 + (35)^3  = 1+ 42875 = 42876`

∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ N

Therefore, the operation * is not associative.

Hence, the operation * is commutative, but not associative. Thus, the correct answer is B.