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A binary operation ⊕ on a set of integers is defined as x ⊕ y = x² × y². Which one of the following statements is TRUE about ⊕?

1. Commutative but not associative
2. Both commutative and associative
3. Associative but not commutative
4. Neither commutative nor associative

Correct answer: Commutative but not associative

Solution

x(+)y = x^2*y^2 = y^2*x^2 so it is commutative. But (x(+)y)(+)z = (x^2 y^2)^2 z^2 = x^4 y^4 z^2 while x(+)(y(+)z) = x^2 (y^2 z^2)^2 = x^2 y^4 z^4, which differ, so it is not associative. Answer: commutative but not associative.

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