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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 days happenings...
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