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Consider a binary operation * on N defined as a * b = a3 + b3. The operation * is
  • a)
    both associative and commutative
  • b)
    commutative, but not associative  
  • c)
    associative, but not commutative
  • d)
    neither commutative nor associative
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Consider a binary operation * on N defined as a * b = a3 + b3. The ope...
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 = (13 + 23) * 3 = 9 * 3 = 93 + 33 = 729 + 27 = 756
1 * (2 * 3) = 1 * (23 + 33) = 1 * (8 + 27) = 1 x 35 = 13 + 353 = 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 (2).
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