A binary operation ⊕ on a set of integers is defined as x ⊕ y = x2 + y2. Which one of the following statements is TRUE about ⊕? A) Commutative but not associative B) Both commutative and associative C) Associative but not commutative D) Neither commutative nor associative

Official answer: A) Commutative but not associative. Explanation: To check commutativity: x ⊕ y = x2 + y2 = y2 + x2 = y ⊕ x Thus, the operation is commutative. To check associativity: (x ⊕ y) ⊕ z = (x2 + y2) ⊕ z = (x2 + y2)2 + z2 x ⊕ (y ⊕ z) = x ⊕ (y2 + z2) = x2 + (y2 + z2)2 Taking a counterexample where x = 1, y = 2, and z = 3: (1 ⊕ 2) ⊕ 3 = (12 + 22)2 + 32 = 52 + 9 = 34 1 ⊕ (2 ⊕ 3) = 12 + (22 + 32)2 = 1 + 132 = 170 Since 34 ≠ 170, the operation is not associative. Therefore, ⊕ is commutative but not associative (Option A).

B) Both commutative and associative

C) Associative but not commutative

D) Neither commutative nor associative

Discrete Mathematics

GATE CSE 2013 · Q1

MCQ1 markEasy

Time00:00

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 $\oplus$ ?

Select one answerSingle correct

Pick an option to check

Solution

Step-by-step verified solution

Official answer: A) Commutative but not associative

Discrete Mathematics

To check commutativity:

x \oplus y = x^2 + y^2 = y^2 + x^2 = y \oplus x

Thus, the operation is commutative.

To check associativity:

(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

Taking a counterexample where $x = 1$ , $y = 2$ , and $z = 3$ :

(1 \oplus 2) \oplus 3 = (1^2 + 2^2)^2 + 3^2 = 5^2 + 9 = 34

1 \oplus (2 \oplus 3) = 1^2 + (2^2 + 3^2)^2 = 1 + 13^2 = 170

Since $34 \neq 170$ , the operation is not associative.

Therefore, $\oplus$ is commutative but not associative (Option A).

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This Discrete Mathematics previous year question is Q1 of the GATE CSE 2013 paper. Find more GATE CSE PYQs on Discrete Mathematics in Engineering Mathematics, or browse all GATE CSE PYQs.

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