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# APPLI DISCRETE STRUC COT 3100

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This 5 page Class Notes was uploaded by Carlo Monahan on Friday September 18, 2015. The Class Notes belongs to COT 3100 at University of Florida taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/206632/cot-3100-university-of-florida in OTHER at University of Florida.

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Date Created: 09/18/15

Name Section Group Number COT 3100 Application of Discrete Structures Spring 2001 No calculators No crib sheets Closed book Closed notes For maximum credit show all work in a clear and organized manner Tuesday April 24 2001 Time 820pm to 10 lOpm l 4 points Give a precise description of Euclid s algorithm for nding the GCD of two integers a and b 2 Convert the following unsigned integers from the given alternative radix expansions to their corresponding decimal expansion Show your hand calculations in the blank area to the right a 3 Points 3FA16 10 b 3 Points 1000 10 3 Find the 6bit two s complement binary representations of the following signed integers given in decimal a 3 points 28 b 3 points 15 4 V39 0 l 00 O 2 points De ne recurrence relation 6 points How many onetoone functions are there from a set with n elements to a set with m elements where m and n are positive integers n g m 6 points How many bit strings of length seven either begin with two 0 s or end with three 1 s or both 6 points What is the coe icient of 11493101 in the expansion of 21 y150 2 points When is a relation R on a set A called antisymmetric 6 points Given a relation R on a set A de ned as R b a is a parent of b with a b E A construct a new relation S in terms off with the property S 11 is a grandparent of c with a c E A 10 4 points Give a precise formal statement of the second principle of mathematical induction ll 4 points Compute the matrix resulting from the following matrix multiplication 3 1 1 3 1 0 2 x 2 0 3 4 O 12 4 points Consider the relation R on a set A 1 2 3 4 R 1 2 1 3 2 4 3 2 3 Is R transitive Give a clear and detailed explanation 13 The relation R on the set ofpositive integers greater than 1 is de ned as R b and b are either both prime or both composite a 9 points Show that R is an equivalence relation b 4 points What are the equivalence classes of 10 and 11 Use set builder notation for your answer c 2 points Prove or disprove The equivalence classes of 10 and 11 partition the set of positive integers greater than 1 EXplicitly state the theorems used in your proof l4 8 points Give a detailed and rigorous proof of the following theorem There is no sequence ofthree consecutive odd numbers 2k 1 2 3 2k 5 with k gt 1that are all prime Hint Use a proof by contradiction and consider the possible values that the three numbers might have mod 3

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