hmwk_1_sol - Discrete Structures Sept 7, 2011 Assignment 1...

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Unformatted text preview: Discrete Structures Sept 7, 2011 Assignment 1 Solutions: Prof. Hopcroft The homework was graded out of 100 points, with 20 points to each problem. 1 Sets Each subsection was worth 5 points. For each of the following, if possible, give 4 examples of elements from each set: a. { i 1 | i N } 1 01 001 0001 b. S 1 = { i 10 i +1 1 | i N } * 101 01001 0010001 10101001 c. S 2 = { 1 }{ i 10 i +1 1 | i N }{ } * { 1 } 1010011 10010001001 1000100001000000001 11011 d. S 1 S 2 1010010001 Note: that is the only element in the intersection. Not even the empty string is in the intersection. 2 Rational Numbers Each section was worth 6 points, with 2 points given for making an attempt on the problems. Prove each of the following: a. Every rational number is a terminating real or a repeating real number. +2 for an answer that address the problem, +2 for mentioning long division, +2 for correctness 1 Proof. Let t be a rational number, then t = p q , where p and q are integers. If p > q , then as a real number t = x.y . Now for this proof we are only interested in whether or not y is terminating or repeating. Hence, we can use t = t- x and only consider t . We can do this because t = 0 .y and if t is a terminating or repeating real number then t is as well. The proof that t is a terminating or repeating real! We note that t is a rational number and can be written as t = n m for integers n < m . To write t as a real number we can perform long division! Long division is a recursive process and involves carrying a remainder at each stage. This remainder must be an integer less than m (or we did long division incorrectly)! Also if we ever see the same remainder, the long division process will just repeat itself and produce a repeating real number. Hence either the long division process ends within m steps and produces a terminating real or sees a duplicate remainder and produces a repeating real!...
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hmwk_1_sol - Discrete Structures Sept 7, 2011 Assignment 1...

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