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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: Neither commutative nor associative

Solution

The operation ⊕ is not commutative because x ⊕ y = x² - y² is not equal to y ⊕ x = y² - x² for all integers x and y. It is also not associative since (x ⊕ y) ⊕ z does not equal x ⊕ (y ⊕ z) when calculated, demonstrating that the grouping of operations affects the result.

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