Suppose U is the power set of the set S = \1,2,3,4,5,6\. For any T U, let |T| denote the number of elements in T and T' denote the complement of T. For any T, R U, let T R be the set of all elements in T which are not in R. Which one of the following is true? A) X U (|X| = |X'|) B) X U \ Y U (|X| = 5, |Y| = 5 and X Y = ) C) X U \ Y U (|X| = 2, |Y| = 3 and X Y = ) D) X U \ Y U (X Y = Y' X')

Official answer: D) X U \ Y U (X Y = Y' X'). Explanation: Let us analyze the given options: - Option (A): X U (|X| = |X'|) is False. If X = , then |X| = 0 while |X'| = |S| = 6 0. - Option (B): X U, Y U (|X| = 5, |Y| = 5 and X Y = ) is False. By the principle of inclusion-exclusion: |X Y| = |X| + |Y| - |X Y| 5 + 5 - 6 = 4 > 0 Hence, X and Y cannot be disjoint. - Option (C): X U, Y U (|X| = 2, |Y| = 3 and X Y = ) is False because not all subsets in U have cardinality 2 and 3. - Option (D): X U, Y U (X Y = Y' X') is True: Using standard set identities: X Y = X Y' Y' X' = Y' (X')' = Y' X = X Y' Since both sides simplify to X Y', the equality holds for all X, Y U. Therefore, (D) is the correct option.

B) X U \ Y U (|X| = 5, |Y| = 5 and X Y = )

C) X U \ Y U (|X| = 2, |Y| = 3 and X Y = )

Engineering Mathematics · Discrete Mathematics

Official IIT answer key · IIT Kanpur · Audited Aug 2026

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GATE CSE 2015 Set 3 Q23 · MCQ · 1 mark

Suppose $U$ is the power set of the set $S = \{1,2,3,4,5,6\}$ . For any $T \in U$ , let $|T|$ denote the number of elements in $T$ and $T^{\prime}$ denote the complement of $T$ . For any $T, R \in U$ , let $T \setminus R$ be the set of all elements in $T$ which are not in $R$ . Which one of the following is true?

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Discrete Mathematics

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In GATE Computer Science, Discrete Mathematics is heavily tested across mathematical logic, set theory, relations & partial orders/lattices, group theory, combinatorics, and graph theory. Questions evaluate formal reasoning, counterexample construction, and structural counting principles. Over recent years, question formats have transitioned from standard single-choice definitions to multi-select properties (MSQs) and exact integer/combinatorial numerical answers (NATs).

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

Engineering Mathematics · Discrete Mathematics

Official IIT answer key · IIT Kanpur · Audited Aug 2026

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GATE CSE 2015 Set 3 Q23 · MCQ · 1 mark

Suppose $U$ is the power set of the set $S = \{1,2,3,4,5,6\}$ . For any $T \in U$ , let $|T|$ denote the number of elements in $T$ and $T^{\prime}$ denote the complement of $T$ . For any $T, R \in U$ , let $T \setminus R$ be the set of all elements in $T$ which are not in $R$ . Which one of the following is true?

Select an option to check

Key concept

Discrete Mathematics

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Examiner trap & misconception

Topic notes

How GATE tests Discrete Mathematics

Full Discrete Mathematics guide →

More on Discrete Mathematics

All 50 questions on Discrete Mathematics →

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

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