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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 $\oplus$?

- Commutative but not associative
- Both commutative and associative
- Associative but not commutative
- Neither commutative nor associative

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Best answer

Answer is (A) Commutative but not associative.

$y \oplus x = y^2 + x^2 = x \oplus y$. Hence, commutative.

$ (x \oplus y) \oplus z = (x^2 + y^2) \oplus z = (x^2 + y^2)^2 + z^2$

$ x \oplus (y \oplus z) = x \oplus (y^2 + z^2) = x^2 + (y^2 + z^2)^2$

So, $( (x \oplus y) \oplus z) \neq (x \oplus (y \oplus z))$, hence not associative.

$y \oplus x = y^2 + x^2 = x \oplus y$. Hence, commutative.

$ (x \oplus y) \oplus z = (x^2 + y^2) \oplus z = (x^2 + y^2)^2 + z^2$

$ x \oplus (y \oplus z) = x \oplus (y^2 + z^2) = x^2 + (y^2 + z^2)^2$

So, $( (x \oplus y) \oplus z) \neq (x \oplus (y \oplus z))$, hence not associative.

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