Assignment1

Assignment1 - (a I am going to go crazy if I see another logical expression(b Studying for exams will not be helpful unless you get enough sleep(c

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Math 240, Fall 2009, Assignment 1 — due Monday September 21. Please explain yourself fully. Writing down a complete truth table does count as an explanation, but may not always be the clearest one! 1. Let p, q be propositions. Construct truth tables for the following four compound expressions: p → ¬ p , p ↔ ¬ p , p ( p q ), ( p q ) ( p q ). 2. Let p, q, r be propositions. Construct the truth tables for ( p q ) r and p ( q r ) (they can go in the same table). 3. Suppose we have a mystery proposition m whose truth table is: p q r m T T T T T T F F T F T F T F F F F T T T F T F T F F T F F F F F (Note: the truth values for m were chosen by tossing a coin 8 times!) (a) Using the operators , , ¬ and the propositions p, q, r , construct an expression which is logically equivalent to m . (b) (peer reviewed question) Describe a way to do this for general m . 4. Translate the following sentences into logical expressions (some of these will require you to use quantifiers).
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Unformatted text preview: (a) I am going to go crazy if I see another logical expression. (b) Studying for exams will not be helpful unless you get enough sleep. (c) Everyone in your class has a cellular phone. (d) Nobody in your class has a satellite phone. (e) There is a person in your class who cannot swim. (f) All students in your class can solve quadratic equations. 5. Establish these logical equivalences, where A is a proposition not involving any quantifiers. • ( ∀ xP ( x )) ∨ A ≡ ∀ x ( P ( x ) ∨ A ) • ( ∃ xP ( x )) ∨ A ≡ ∃ x ( P ( x ) ∨ A ) 6. Determine the truth value of these statements if the universe of discourse for all variables consists of all real numbers. Explain. • ∀ x ∃ y ( x 2 = y ) • ∃ x ∀ y ( xy = 0) • ∃ x ∃ y ( x + 2 y = 2 ∧ 2 x + 4 y = 5)...
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This note was uploaded on 10/03/2010 for the course MATH 240 taught by Professor Szabo during the Spring '08 term at McGill.

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