Consider the operators and defined by a b = a + 2b, a b = ab, for positive integers. Which of the following statements is/are TRUE? A) Operator obeys the associative law B) Operator obeys the associative law C) Operator over the operator obeys the distributive law D) Operator over the operator obeys the distributive law

Official answer: B) Operator obeys the associative law; D) Operator over the operator obeys the distributive law. Explanation: Let the two binary operations on positive integers be: a b = a + 2b a b = ab Let us evaluate each statement: - Option (A): Associativity of requires (a b) c = a (b c). (a b) c = (a + 2b) c = (a + 2b) + 2c = a + 2b + 2c a (b c) = a (b + 2c) = a + 2(b + 2c) = a + 2b + 4c Since $a + 2b + 2c e a + 2b + 4c for positive integers, $ is not associative. Thus, (A) is FALSE. - Option (B): Associativity of requires (a b) c = a (b c). (a b) c = (ab) c = (ab)c = abc a (b c) = a (bc) = a(bc) = abc Since multiplication is associative, (a b) c = a (b c). Thus, (B) is TRUE. - Option (C): Distributivity of over requires a (b c) = (a b) (a c). LHS = a (bc) = a + 2bc RHS = (a b) (a c) = (a + 2b)(a + 2c) = a2 + 2ac + 2ab + 4bc Since $LHS e RHS$ in general, (C) is FALSE. - Option (D): Distributivity of over requires a (b c) = (a b) (a c). LHS = a (b + 2c) = a(b + 2c) = ab + 2ac RHS = (ab) (ac) = ab + 2(ac) = ab + 2ac Since LHS = RHS, distributes over . Thus, (D) is TRUE. Therefore, the correct statements are B and D.

A) Operator obeys the associative law

B) Operator obeys the associative law

C) Operator over the operator obeys the distributive law

D) Operator over the operator obeys the distributive law

Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IISc Bangalore · Audited Aug 2026

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GATE CSE 2024 Set 1 Q52 · MSQ · 2 marks

Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?

Select one or more options

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Sets, Relations, Lattices and Groups

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How GATE tests Sets, Relations, Lattices and Groups

In GATE CS, questions on sets, relations, and functions test formal mathematical rigor through algebraic verification of binary relation properties (reflexivity, symmetry, transitivity), set-theoretic properties of direct and inverse images of functions, and state-reachability or partition analysis on integer domain functions. The questions range from conceptual 1-mark MCQs on standard definitions and counterexamples to 2-mark algebraic or NAT problem-solving items involving equivalence classes and modular cycles.

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Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IISc Bangalore · Audited Aug 2026

Take topic test

GATE CSE 2024 Set 1 Q52 · MSQ · 2 marks

Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?

Select one or more options

Key concept

Sets, Relations, Lattices and Groups

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Sets, Relations, Lattices and Groups

Full Sets, Relations, Lattices and Groups guide →

More on Sets, Relations, Lattices and Groups

All 11 questions on Sets, Relations, Lattices and Groups →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?

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Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?

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Consider the operators ◊\Diamond◊ and □\square□ defined by a◊b=a+2ba \Diamond b = a + 2ba◊b=a+2b, a□b=aba \square b = aba□b=ab, for positive integers. Which of the following statements is/are TRUE?

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Consider the operators ◊\Diamond◊ and □\square□ defined by a◊b=a+2ba \Diamond b = a + 2ba◊b=a+2b, a□b=aba \square b = aba□b=ab, for positive integers. Which of the following statements is/are TRUE?

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Check you can still solve Sets, Relations, Lattices and Groups under a clock

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Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?

Consider the operators $\Diamond$ and $\square$ defined by $a \Diamond b = a + 2b$ , $a \square b = ab$ , for positive integers. Which of the following statements is/are TRUE?