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Unformatted text preview: MACM 101 — Discrete Mathematics I Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) Reminder: the work you submit must be your own. Any collabora tion and consulting outside resourses must be explicitely mentioned on your submission. Please, use a pen. 30 points will be taken off for pencil written work. 1. Determine the truth value of each of these statements if the universe of each variable consists of (i) all real numbers, (ii) all integers. (a) ∃ x ∃ y ( x + y 6 = y + x ) (b) ∀ x ∃ y ( x + y = 2 ∧ 2 x y = 2) Solution (a) Formally negating the statement we get ∀ x ∀ y ( x + y = y + x ) , which is the law of commutativity of addition. Thus statement (a) is false in both universes, because addition is com mutative and for any x,y we have x + y = y + x . (b) The statement is false in both universes. To prove it we need to prove that negation of this statement is true. ¬ ( ∀ x ∃ y ( x + y = 2 ∧ 2 x y = 2)) ⇔ ∃ x ∀ y x + y 6 = 2 ∨ 2 x y 6 = 2) Let us assign x = 2 and then the quantified predicate turns into 2 + y 6 = 2 ∨ 4 y 6 = 2 ⇔ y 6 = 0 ∨ y 6 = 2 . We see that the quantified statement is true for all y and thus we have the negation of statement (b) proven. 1 2. Use predicates and quantifiers to express this statement “There is a man who has visited some park in every province of Canada” Solution Let V ( x,y ), where x is a person and y is a park be a predicate “Person x visited park y ”....
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 Spring '09
 jcliu
 Math, Logic, quantifier

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