cps102-hw2

cps102-hw2 - 2! + ... + n n ! = ( n + 1)!-1 whenever n is a...

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CPS 102: Discrete Mathematics Instructor: Bruce Maggs Assignment 2 Due: Monday, October 3th, 2011 1 (15 points) Prove that 1 / 3 is a recurring decimal number (in other words, it does not have a finite decimal representation). Hint: Use induction on the number of digits in any representation. 2 (4 points) Prove that if n is an integer and 3 n + 2 is even, then n is even using an indirect proof (an indirect proof proves an argument of the form, if p is true then q is true , by proving the contrapositive - if q is false , then p is false ) a proof by contradiction 3 (6 points) Prove that except for 3, 5, 7, no three consecutive odd positive integers are prime. 4 (15 points) Prove that 2 is an irrational number. Hint : Any rational number can be represented using integers p and q , in the form p q such that q 6 = 0 and with GCD ( p,q ) = 1. 5 (5 points) Prove that there are infinitely many primes. 6 (5 points) Prove or disprove: If a , b , and m are positive integers, then ( a mod m ) + ( b mod m ) = ( a + b ) mod m 1
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7 (3 points) Using mathematical induction prove that 1 · 1! + 2
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Unformatted text preview: 2! + ... + n n ! = ( n + 1)!-1 whenever n is a positive integer. 8 (3 points) Use mathematical induction to prove that 1 2 3+2 3 4+ ... + n ( n +1) ( n +2) = n ( n +1)( n +2)( n +3) / 4 9 (4 points) Use mathematical induction to prove that a set with n elements has n ( n-1)( n-2) / 6 subsets containing exactly three elements whenever n is an integer greater than or equal to 3 10 (5 points) Find the aw with the following proof that a n = 1 for all nonnegative integers n , whenever a is a nonzero real number. Basic Step : a = 1 is true by the denition of a . Inductive Step : Assume that a k = 1 for all nonnegative integers k with k n . Then note that a n +1 = a n a n a n-1 = 1 1 1 = 1 11 (15 points) Prove that for any integer n a 2 n 2 n grid with one corner square removed can be covered with L shaped tiles made up of 3 squares. 2...
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cps102-hw2 - 2! + ... + n n ! = ( n + 1)!-1 whenever n is a...

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