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hw2 - If it is hot outside and the air conditioner is not...

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CprE 310 Theoretical Foundations of Computer Engg, Spring 08 Homework 2 Due: 7 Feb 2008, In class Topics: Proofs, Propositions, Sets and Functions Reading: Sections 1.1, 1.2, 1.6, 1.7, 2.1, 2.2, 2.3 Problem 2.1 Use truth tables to show that the following pairs of propositions are equiv- alent. a. p q and ¬ q → ¬ p b. p ( q r ) and ( p q ) ( p r ) Problem 2.2 A propositional formula F is said to be in Conjunctive Normal Form (CNF) if F is an AND of conjuncts C 1 , C 2 . . . , where each conjunct C i is an OR of literals and each literal is a symbol representing a proposition or its negation. For example, the following formulas are in CNF: ( b ∨ ¬ c ) ( a d ) a ∨ ¬ b The following are not: ( b c ) d a b Question: Convert the following propositional formula into an equivalent one which is in CNF. ( p q ) r
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CprE 310 Homework 2 7 Feb 2008, In class Problem 2.3 Let p be the proposition “the Air Conditioner is not working”, q be “it is hot outside”, r “it is hot inside” and s “the alternate cooling system is in place”. Express the following sentence as a propositional formula.
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Unformatted text preview: If it is hot outside, and the air conditioner is not working, then it is hot inside unless the alternate cooling system is in place . Problem 2.4 Let A, B, C be any three sets and let U denote the universe of all elements. Prove the following: ( B-A ) ∪ ( C-A ) = ( B ∪ C )-A Problem 2.5 Let R denote the set of real numbers. For each of the following conditions, give an example of a function from R to R that satis±es the condition. Explain your answer brie²y. a. The function is one-one but not onto b. The function is onto but not one-one c. The function is both one-one and onto d. The function is neither one-one nor onto Problem 2.6 Let Z denote the set of integers and E denote the set of even numbers, i.e. E = { 2 , 4 , 6 , 8 , . . . } . Demonstrate a function g that is a bijection from Z to E , and prove that the function is a bijection. 2...
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