hw0sol - Problem Set 0 Solutions CS373 - Spring 2011 Due:...

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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 non-empty 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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hw0sol - Problem Set 0 Solutions CS373 - Spring 2011 Due:...

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