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Unformatted text preview: 1 Solutions Problem 1.1 Indicate which of the following sentences are propositions. a. 1,024 is the smallest fourdigit number that is perfect square. b. She is a mathematics major. c. 128 = 2 6 d. x = 2 6 . Solution. a. Proposition with truth value (T). Note that 32 2 = 1024 and 31 2 = 961 . b. Not a proposition since the truth or falsity of the sentence depends on the reference for the pronoun ”she.” For some values of ”she” the statement is true; for others it is false. c. A proposition with truth value (F). d. Not a proposition Problem 1.2 Consider the propositions: p: Juan is a math major. q: Juan is a computer science major. Use symbolic connectives to represent the proposition ”Juan is a math major but not a computer science major.” Solution. p ∧ ∼ q Problem 1.3 In the following sentence is the word ”or” used in its inclusive or exclusive sense? ”A team wins the playoffs if it wins two games in a row or a total of three games.” Solution. Exclusive 2 Problem 1.4 Write the truth table for the proposition: ( p ∨ ( ∼ p ∨ q )) ∧ ∼ ( q ∧ ∼ r ) . Solution. Let s = ( p ∨ ( ∼ p ∨ q )) ∧ ∼ ( q ∧ ∼ r ) . p q r ∼ p ∼ r ∼ p ∨ q q ∧ ∼ r ∼ ( q ∧ ∼ r ) p ∨ ( ∼ p ∨ q ) s T T T F F T F T T T T T F F T T T F T F T F T F F F F T T T T F F F T F F T T T F T T T F T F T T T F T F T T T T F T F F F T T F T F T T T F F F T T T F T T T Problem 1.5 Let t be a tautology. Show that p ∨ t ≡ t. Solution. p t p ∨ t T T T F T T Problem 1.6 Let c be a contradiction. Show that p ∨ c ≡ p. Solution. p c p ∨ c T F T F F F Problem 1.7 Show that ( r ∨ p ) ∧ (( ∼ r ∨ ( p ∧ q )) ∧ ( r ∨ q )) ≡ p ∧ q. 3 Solution. Let s = ( ∼ r ∨ ( p ∧ q )) ∧ ( r ∨ q ) . p q r r ∨ p r ∨ q ∼ r ∨ ( p ∧ q ) s ( r ∨ p ) ∧ s p ∧ q T T T T T T T T T T T F T T T T T T T F T T T F F F F T F F T F T F F F F T T T T F F F F F T F F T T F F F F F T T T F F F F F F F F F T F F F Problem 1.8 Use De Morgan’s laws to write the negation for the proposition:”This com puter program has a logical error in the first ten lines or it is being run with an incomplete data set.” Solution. ”This computer program is error free in the first ten lines and it is being run with complete data.” Problem 1.9 Use De Morgan’s laws to write the negation for the proposition:”The dollar is at an alltime high and the stock market is at a record low.” Solution. ”The dollar is not at an alltime high or the stock market is not at a record low.” Problem 1.10 Assume x ∈ R . Use De Morgan’s laws to write the negation for the proposition:0 ≥ x > 5 . Solution. x ≤  5 or x > . 4 Problem 1.11 Show that the proposition s = ( p ∧ q ) ∨ ( ∼ p ∨ ( p ∧ ∼ q )) is a tautology....
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 Spring '08
 RAMASWAMY
 Math, Natural number, Prime number, Rational number

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