GATE CS 2015 Set 1 — Question 20
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Question
For a set , the power set of is denoted by . If , which of the following options are TRUE?I.
II.
III.
IV.
II.
III.
IV.
Correct answer
(C) I, II and III only
Solution
Given . The power set is the set of all subsets of .
- Statement I: . Since the empty set is a subset of any set , it must be an element of the power set . Thus, I is TRUE.
- Statement II: . The empty set is a subset of every set, including . Thus, II is TRUE.
- Statement III: . For to be an element of , it must be a subset of . Since and , the set is indeed a subset of . Thus, III is TRUE.
- Statement IV: . For this to be true, every element of must be an element of . This means and . However, and are elements of , not subsets of . The subsets of are elements of (e.g., and ). Since and , IV is FALSE.
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