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Unformatted text preview: Math 230 E Fall 2011 Homework 2 Solutions 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 ∧ , ⊕ , ¬ . Some trialanderror is involved in this, but using Venn diagrams and the idea that x ⊕ y is the “same” as x Δ y can help a lot. Note that there are infinitely many correct answers, so yours may differ from mine. First, I claim that x ⊕ y = ¬ ( x ⇔ y ) = ¬ (( x ⇒ y ) ∧ ( y ⇒ x )) = ¬ ( ¬ ( x ∧¬ y ) ∧¬ ( y ∧¬ x )) = ( x ∧¬ y ) ∨ ( y ∧¬ x ) = various other things. To prove this I will use a truth table: x y x ⇔ y ¬ ( x ⇔ y ) x ⊕ y 1 1 1 1 0 1 1 0 1 1 1 0 0 1 Note that the last two columns are equal. The very last column was given to us by definition . Second, I claim that x ∨ y = ( x ⊕ y ) ⊕ ( x ∧ y ) = various other things. To prove this we again use a truth table: x y x ⊕ y x ∧ y ( x ⊕ y ) ⊕ ( x ∧ y ) x ∨ y 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 Note that the last two columns are equal. [FYI: The structure ( { , 1 } , ∧ , ∨ , ¬ ) is called a “Boolean algebra” — as we know — and the structure ( { , 1 } , ∧ , ⊕ , ¬ ) is called a “Boolean ring”. In a Boolean ring we can really think of ∧ as “multiplication mod 2 ” and ⊕ as “addition mod 2 ”.] Problem 2. Given two subsets S,T ⊆ U of some universal set, the statement “ S = T ” means exactly that “ ∀ x ∈ U,x ∈ S ⇔ x...
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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.
 Fall '11
 Armstrong
 Math

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