Question

Consider a binary operation *on N defined as $a*b=a^3+b^3.$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

Answer: (B)

Is * commutative but not associative?

Answer 2

On N , the operation * is defined as $a*b=a^3+b^3.$
For, a, b, ∈ N , we have: $a*b=a^3+b^3=b^3+a^3=b*a$ [Addition is commutative in N]
Therefore, the operation * is commutative. It can be observed that:
$(1*2)*3=(1^3+2^3)*3=9*3$= $9^3+3^3=756$
$1*(2*3)=1*(2^3+3^3)=1*(8+27)=1*35=1^3+35^3=1+35^3=42876$
∴ (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.