MIT6_042JS10_lec05_sol

# MIT6_042JS10_lec05_sol - Massachusetts Institute of...

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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 In-Class 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 take-away 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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MIT6_042JS10_lec05_sol - Massachusetts Institute of...

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