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Unformatted text preview: Problem Set 0 Solutions CS373  Spring 2011 Due: 1. True/false [ Category : Notation, Points : 10] Answer each of the following with true or false . Follow the notations in Sipser . We use { ... } to represent sets (not, for example, multisets). The symbol P ( S ) denotes the set of all subsets of S (the power set of S ). The symbol A represents an arbitrary nonempty set. D1) { } True D2) {{ }} True, the empty set is a subset of all sets. D3) A P ( P ( A ) \ ) False, the individual elements of A are not elements of P ( A ) , so A is not an element of P ( P ( A ) \ ) D4) A P ( P ( A ) \ A ) False, see above D5)  P ( P ( P ( A ) \ A ))  = 0 False, the powerset of a set always contains the empty set as a member. D6)  P ( P ( A )) \ ( { A } A )  6 = 0 True, see above D7) P ( P ( A )) P ( P ( P ( A ))) False, P ( P ( A )) contains { A } as an element, while P ( P ( P ( A ))) does not contain { A } as an element. D8) P ( P ( )) P ( P ( P ( ))) True, P ( P ( )) = { , { }} , and P ( P ( P ( ))) = { , { } , {{ }} , { , { }}} . D9) { } P ( P ( A ) \ ) True, a powerset always contains an empty set as an element, so the set containing just the empty set is always a subset of a powerset. D10) A = \ A False, all elements of \ A must be elements of , which has no elements....
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 Spring '08
 Viswanathan,M

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