A binary operation \(\oplus\) on a set of integers is defined as \(x \oplus y…
2013
A binary operation \(\oplus\) on a set of integers is defined as \(x \oplus y = x^{2}+y^{2}\) . Which one of the following statements is TRUE about \(\oplus\)?
- A.
Commutative but not associative
- B.
Both commutative and associative
- C.
Associative but not commutative
- D.
Neither commutative nor associative
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Correct answer: A
Key idea: check commutativity and associativity for the operation x ⊕ y = x² + y².
Commutativity:
x ⊕ y = x² + y² = y² + x² = y ⊕ x, so the operation is commutative for all integers x and y.
Associativity:
Compute (x ⊕ y) ⊕ z = (x² + y²)² + z² and x ⊕ (y ⊕ z) = x² + (y² + z²)². These expressions are not equal in general.
Example: x = 1, y = 2, z = 3 gives (1 ⊕ 2) ⊕ 3 = (1+4)² + 9 = 25 + 9 = 34, while 1 ⊕ (2 ⊕ 3) = 1 + (4+9)² = 1 + 169 = 170. Since 34 ≠ 170, associativity fails.
Conclusion: the operation is commutative but not associative, so the correct statement is: "Commutative but not associative."
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