60 votes 60 votes Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true? $P(P(S)) = P(S)$ $P(S) ∩ P(P(S)) = \{ Ø \}$ $P(S) ∩ S = P(S)$ $S ∉ P(S)$ Set Theory & Algebra gatecse-2000 set-theory&algebra easy set-theory + – Kathleen asked Sep 14, 2014 edited May 29, 2018 by Arjun Kathleen 13.4k views answer comment Share Follow See all 6 Comments See all 6 6 Comments reply Show 3 previous comments nishant_magarde commented Dec 7, 2019 reply Follow Share one more option should be there. E. None of these 0 votes 0 votes smsubham commented Dec 26, 2019 reply Follow Share https://math.stackexchange.com/questions/2836572/can-a-set-and-its-powerset-have-anything-in-common?fbclid=IwAR2cWb5g3nNf3hY-0zD8bvdK1M7tKOsHleiRmxlhmW8Bv7Fd0VVMl1fYNRQ 0 votes 0 votes Deepak Poonia commented Apr 23, 2022 reply Follow Share 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 4 votes 4 votes Please log in or register to add a comment.
3 votes 3 votes 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. worst_engineer answered Jul 19, 2015 worst_engineer comment Share Follow See all 2 Comments See all 2 2 Comments reply srestha commented Sep 17, 2015 reply Follow Share yes u r correct 0 votes 0 votes vnc commented Nov 2, 2015 reply Follow Share 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). 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes 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} 😊😊😊😊😊😊😊😊😊😊😊😊 akshay_123 answered Sep 14, 2023 akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.