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230hw2 - Math 230 E Homework 2 Fall 2011 Drew Armstrong...

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Math 230 E Fall 2011 Homework 2 Drew Armstrong Problem 1. Let x y denote the XOR (exclusive or) function, which is defined by: 1 1 = 0 , 1 0 = 1 , 0 1 = 1 , 0 0 = 0 . Express x y in terms of the basic operations , , ¬ . Express the usual “inclusive or” x y in terms of the operations , , ¬ . [Boolean logic could be based on either of the triples ( , , ¬ ) or ( , , ¬ ) . Once upon a time somebody made a choice.] Problem 2. Given two subsets S, T U of some universal set, the statement “ S = T means exactly that “ x U, x S x T ”. Define the symmetric di ff erence by S Δ T := ( S T c ) ( T S c ). Verify the following logical equivalence: S = T ” = “ S Δ T = . (Hint: Use the principle ¬ ( x U, P ( x )) = ( x U, ¬ P ( x )) and de Morgan’s Law.) [Food for thought: What is the relationship between x y and S Δ T ?] Problem 3. For this problem you may assume de Morgan’s Laws: ¬ ( P 1 P 2 · · · P k ) = ¬ P 1 ¬ P 2 · · · ¬ P k ¬ ( P
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