For a set 𝐴, the power set of 𝐴 is denoted by 2𝐴. If 𝐴 = {5,{6},{7}},…
2015
For a set 𝐴, the power set of 𝐴 is denoted by 2𝐴. If 𝐴 = {5,{6},{7}}, which of the following options are TRUE?
I. ∅ ∈ 2𝐴
II. ∅ ⊆ 2𝐴
III. {5,{6}} ∈ 2𝐴
IV. {5,{6}} ⊆ 2𝐴
Answer: C. I, II and III only — Key idea: the power set 2A consists of all subsets of A. ∅ ∈ 2A because the empty set is a subset of every set (so it is one of the subsets of A). ∅ ⊆ 2A…
- A.
I and III only
- B.
II and III only
- C.
I, II and III only
- D.
I, II and IV only
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Correct answer: C
Key idea: the power set 2A consists of all subsets of A.
∅ ∈ 2A because the empty set is a subset of every set (so it is one of the subsets of A).
∅ ⊆ 2A because the empty set is a subset of every set.
{5,{6}} ∈ 2A because {5,{6}} is a subset of A (both 5 and {6} are elements of A), so that subset is one element of the power set.
{5,{6}} ⊆ 2A is false because its elements (5 and {6}) are not subsets of A and therefore are not elements of 2A.
Conclusion: I, II and III are true; IV is false. So the correct selection is I, II and III only.
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