GATE CSE 2000 | Question: 2.6

Question

Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true?

• A. $P(P(S)) = P(S)$
• B. $P(S) ∩ P(P(S)) = \{ Ø \}$
• C. $P(S) ∩ S = P(S)$
• D. $S ∉ P(S)$

Comments

one more option should be there.

E. None of these

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https://math.stackexchange.com/questions/2836572/can-a-set-and-its-powerset-have-anything-in-common?fbclid=IwAR2cWb5g3nNf3hY-0zD8bvdK1M7tKOsHleiRmxlhmW8Bv7Fd0VVMl1fYNRQ

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Option A,C,D are $\color{red}{\text{NEVER true}}$ for any set $S.$

There is NO set $S(finite \,\,or\,\,\,infinite)$, for which statements in Option A,C,D are true. 

Statement in option B is NOT always true for every set $S,$ but it is true if set $S$ contains simple elements i.e. elements which are not set.

$\color{red}{\text{Find Video Solution Below:}}$

Detailed Video Solution

Answer 1

Let S = {1,2}

P(S) = {{},(1,1),(1,2),(2,1),(2,2)}

P(P(S)) = {{} , (1,1,1,1) , (1,1,1,2) , (1,1,2,1) , (1,1,2,2)...................................... }

So , intersection of P(S) and P(P(S)) = phi

 

Please correct me , if my understanding is wrong.

Comments

yes u r correct

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Your answer is correct but you explanation is wrong.let s{1,2} then p(s)={{},{1},{2},{1,2}} including {phi} there will 4 element(2^2).

Answer 2

ϕ always present in any power set of a set and ϕ is the only common element between P(S) and P(P(S))

Therefore, 

P(S)  ∩  P(P(S) = {ϕ}

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