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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring 10 : Mathematics for Computer Science February 12 Prof. Albert R. Meyer revised February 11, 2010, 172 minutes Solutions to InClass Problems Week 2, Fri. Problem 1. Set Formulas and Propositional Formulas. (a) Verify that the propositional formula ( P AND NOT ( Q )) OR ( P AND Q ) is equivalent to P . Solution. There is a simple verification by truth table with 4 rows which we omit. There is also a simple cases argument: if Q is T , then the formula simplifies to ( P AND F ) OR ( P AND T ) which further simplifies to ( F OR P ) which is equivalent to P . Otherwise, if Q is F , then the formula simplifies to ( P AND T ) OR ( P AND F ) which is likewise equivalent to P . (b) Use part ( a ) to prove that A = ( A B ) ( A B ) for any sets, A,B , where A B ::= { a A  a / B } . Solution. We need only show that the two sets have the same elements, that is x is in one set iff x is in the other set, for any x . Let P be x A and Q be x B . Then x ( A B ) ( A B ) iff x ( A B ) OR x ( A B ) (by def of ) iff ( x A AND NOT ( x B )) OR ( x A AND x B ) (by def of and ) iff ( P AND NOT ( Q )) OR ( P AND Q ) (by def of P and Q ) iff P (by part ( a )) iff x A (by def of P ) . Problem 2. Subset takeaway 1 is a two player game involving a fixed finite set, A . Players alternately choose nonempty subsets of A with the conditions that a player may not choose the whole set...
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 Spring '11
 Prof.AlbertR.Meyer
 Computer Science

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