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hw3 - guage is not ”C” CprE 310 Homework 3 14 Feb 2008...

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CprE 310 Theoretical Foundations of Computer Engg, Spring 08 Homework 3 Due: 14 Feb 2008, In class Topics: Predicates, Proof by Induction Reading: Sections 1.3, 4.1 Problem 3.1 Let P ( x ) be the predicate ” x = x 2 ”. If Z denotes the set of integers, then what are the truth values of the following logical expressions? a. P (0) b. P (2) c. ( x Z )[ P ( x )] d. ( x Z )[ P ( x )] Problem 3.2 Translate the following statements into logical expressions by defining ap- propriate sets and using appropriate quantifiers. a. Everyone in the class has a cell phone. b. There is someone in the class whose birthday is on Feb 29th. c. There is someone in the class who wasn’t born in February. d. No student in the class was born in February. e. There are some students in the class whose favorite programming lan-
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Unformatted text preview: guage is not ”C”. CprE 310 Homework 3 14 Feb 2008, In class Problem 3.3 Prove the following for every integer n > 0. 1 2 + 2 2 + . . . + n 2 = n ( n + 1)(2 n + 1) 6 Problem 3.4 Prove the following for any integer n > 1. 1 1 · 2 + 1 2 · 3 + ··· + 1 ( n-1) n = n-1 n Problem 3.5 For any real number x > 0 and integer n ≥ 2, prove the following. (1 + x ) n > 1 + nx Problem 3.6 For any ±nite set S , let | S | denote the number of elements in S . For any positive integer n , for any collection of n ±nite sets S 1 , S 2 , . . . , S n , prove the following. | S 1 ∪ S 2 ∪ ··· ∪ S n | ≤ | S 1 | + | S 2 | + . . . + | S n | 2...
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