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

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Unformatted text preview: 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 , ⊕ 1 = 1 , ⊕ 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 difference 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:...
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.

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