Let X be a set and 2X denote the powerset of X. Define a binary operation on 2X as follows: A B = (A - B) (B - A) \ . Let H = (2X, ). Which of the following statements about H is/are correct? A) H is a group. B) Every element in H has an inverse, but H is NOT a group. C) For every A 2X, the inverse of A is the complement of A. D) For every A 2X, the inverse of A is A.

Official answer: A) H is a group.; D) For every A 2X, the inverse of A is A.. Explanation: To determine whether H = (2X, ) is a group and find the inverses of its elements, we check the group axioms for the symmetric difference operation on the power set 2X: 1. Closure: For any A, B 2X, the symmetric difference A B = (A - B) (B - A) is a subset of X, so A B 2X. 2. Associativity: The symmetric difference of sets is associative, i.e., (A B) C = A (B C) for all A, B, C 2X. 3. Identity Element: The empty set 2X acts as the identity element because: A = (A - ) ( - A) = A = A 4. Inverse Element: For any set A 2X: A A = (A - A) (A - A) = = Since A A = (the identity element), every element A is its own inverse. Thus: - (A) is correct: H is a group (specifically, a Boolean group). - (B) is incorrect: H is indeed a group. - (C) is incorrect: The inverse of A is A, not its complement Ac (since A Ac = X in general). - (D) is correct: For every A 2X, the inverse of A is A. Hence, the correct options are A and D.

B) Every element in H has an inverse, but H is NOT a group.

C) For every A 2X, the inverse of A is the complement of A.

D) For every A 2X, the inverse of A is A.

Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IIT Kanpur · Audited Aug 2026

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GATE CSE 2023 Q51 · MSQ · 2 marks

Let $X$ be a set and $2^X$ denote the powerset of $X$ . Define a binary operation $\Delta$ on $2^X$ as follows: $A \Delta B = (A - B) \cup (B - A) \ .$ Let $H = (2^X, \Delta)$ . Which of the following statements about $H$ is/are correct?

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Sets, Relations, Lattices and Groups

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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 · IIT Kanpur · Audited Aug 2026

Take topic test

GATE CSE 2023 Q51 · MSQ · 2 marks

Let $X$ be a set and $2^X$ denote the powerset of $X$ . Define a binary operation $\Delta$ on $2^X$ as follows: $A \Delta B = (A - B) \cup (B - A) \ .$ Let $H = (2^X, \Delta)$ . Which of the following statements about $H$ is/are correct?

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.

Let $X$ be a set and $2^X$ denote the powerset of $X$ . Define a binary operation $\Delta$ on $2^X$ as follows: $A \Delta B = (A - B) \cup (B - A) \ .$ Let $H = (2^X, \Delta)$ . Which of the following statements about $H$ is/are correct?

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Let $X$ be a set and $2^X$ denote the powerset of $X$ . Define a binary operation $\Delta$ on $2^X$ as follows: $A \Delta B = (A - B) \cup (B - A) \ .$ Let $H = (2^X, \Delta)$ . Which of the following statements about $H$ is/are correct?

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