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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science February 1, 2005 Srini Devadas and Eric Lehman Problem Set 1 Solutions Due: Monday, February 7 at 9 PM Problem 1. The connectives ∧ (and), ∨ (or), and ⇒ (implies) come often not only in com puter programs, but also everyday speech. But devices that compute the nand operation are preferable in computer chip designs. Here is the truth table for nand: P Q T T F T F T F T T F F T P nand Q For each of the following expressions, find an equivalent expression using only nand and ¬ (not). (a) A ∧ B Solution. ¬ ( A nand B ) (b) A ∨ B Solution. ( ¬ A ) nand ( ¬ B ) (c) A B ⇒ Solution. A nand ( ¬ B ) Problem 2. A selfproclaimed “great logician” has invented a new quantifier, on par with ∃ (“there exists”) and ∀ (“for all”). The new quantifier is symbolized by U and read “there exists a unique”. The proposition Ux P ( x ) is true iff there is exactly one x for which P ( x ) is true. The logician has noted, “There used to be two quantifiers, but now there are three! I have extended the whole field of mathematics by 50%!” (a) Write a proposition equivalent to Ux P ( x ) using only the ∃ quantifier, =, and logical connectives. Solution. x ( P ( x ) ∧ ¬ ( ∃ y ( ¬ ( x = y ) ∧ P ( y ))) ∃ 2 Problem Set 1 (b) Write a proposition equivalent to Ux P ( x ) using only the ∀ quantifier, =, and logical connectives. Solution. = y ∨ ¬ P ( y ))) ¬∀ x ( ¬ P ( x ) ∨ ¬∀ y ( x Problem 3. A media tycoon has an idea for an allnews television network called LNN: The Logic News Network. Each segment will begin with the definition of some relevant sets and predicates. The day’s happenings...
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 Spring '11
 Dr.EricLehman
 Computer Science, Logic

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