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Yellow MT_sln

# Yellow MT_sln - Prof Bjorn Poonen MATH 55 MIDTERM...

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Prof. Bjorn Poonen October 16, 2001 MATH 55 MIDTERM SOLUTIONS (yellow) (1) (5 pts. each) For each of (a)-(g) below: If the proposition is true, write TRUE. If the proposition is false, write FALSE. (Please do not use the abbreviations T and F, since in handwriting they are sometimes indistiguishable.) No explanations are required in this problem. (a) The set { 0 , 1 } * of bit strings of finite length is a countable set. TRUE, because one can list all the bit strings in a sequence, by listing all bit strings of length zero, then all of length one, and so on: λ, 0 , 1 , 00 , 01 , 10 , 11 , . . . Alternatively, { 0 , 1 } * can be viewed as a subset of the set of all finite strings of typewriter symbols, which was shown in class to be countable. (b) The proposition x y ( x y ) is true, when the universe of discourse is the set of natural numbers. FALSE. In words, this says “There is a natural number x that is greater than or equal to all natural numbers y ,” which is false. Alternatively, we can prove the negation, x y ( x < y ), as follows. Suppose that x is a natural number. Then there exists a natural number y such that x < y , namely y = x + 1. (c) The numbers 34 , 35 , 36 are pairwise relatively prime. FALSE, because gcd(34 , 36) = 2. (d) There are infinitely many integers x satisfying both x 12 (mod 99) and x 16 (mod 100) . (Hint: you don’t need to solve this system.) TRUE. Since gcd(99 ,

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Yellow MT_sln - Prof Bjorn Poonen MATH 55 MIDTERM...

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