GATE 2015 CS (CS3) – Question 12

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple choice

U is the power set of S={1,2,3,4,5,6}. |T| is cardinality, T′ complement in S, and T∖R set difference. Which is true?

$\forall X\in U\ (|X|=|X\prime|)$
$\exists X,Y\in U\ (|X|=5,|Y|=5,X\cap Y=\varnothing)$
$\forall X,Y\in U\ (|X|=2,|Y|=3,X\cap Y=\varnothing)$
$\forall X,Y\in U\ (X\setminus Y=Y\prime\setminus X\prime)$

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Correct answer: (D) $\forall X,Y\in U\ (X\setminus Y=Y\prime\setminus X\prime)$

Explanation

By element membership, $X\setminus Y=X\cap Y\prime$. Also $Y\prime\setminus X\prime=Y\prime\cap X$, the same set, so D holds for every pair. A fails for empty X; B is impossible because two disjoint five-element subsets cannot fit in six elements; C is not universal.

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