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

(B) Is * commutative but not associative

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The binary operation * on set N is defined as

a * b = a³ +b³

Now, b * a = b³ + a³ = a³ +b³.

⇒ a * b = b * a.

∴ The operation * is commutative.

Further, a * (b * c) = a * (b³ + c³) = a³ + (b³ + c³)³

and (a * b) * c = (a³ + b³) * c = (a³ +b³)³ + c³

⇒ a * (b * c) ≠ (a * b) * c

∴ This operation is not associative.

Thus, this operation is commutative but not associative.