# hw0sol - Problem Set 0 Solutions CS373 Spring 2011 Due 1...

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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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## This note was uploaded on 04/28/2011 for the course CS 373 taught by Professor Viswanathan,m during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw0sol - Problem Set 0 Solutions CS373 Spring 2011 Due 1...

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