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प्रश्न
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?
उत्तर
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.
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