On the integers define x y=x2+y2. Which property holds?

Associative but not commutative

GATE 2013 CS (CS1) – Question 11

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 1 mark · Multiple choice

On the integers define $x\oplus y=x^2+y^2$. Which property holds?

Commutative but not associative
Commutative and associative
Associative but not commutative
Neither

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Correct answer: (A) Commutative but not associative

Explanation

Swapping x,y leaves x²+y² unchanged, so the operation is commutative. For associativity take x=1,y=0,z=0: (1⊕0)⊕0=1 and 1⊕(0⊕0)=1 does not distinguish; instead use x=2,y=1,z=0. The left side is 25 and the right side 5, proving nonassociativity. Thus A.

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