pset1_soln

# pset1_soln - 6.042/18.062J Mathematics for Computer Science...

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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 self-proclaimed “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 all-news 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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pset1_soln - 6.042/18.062J Mathematics for Computer Science...

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