f2006Soln - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F — Discrete Mathematics Final Examination December 18, 2006 SOLUTIONS Duration: 2.5 hours This is a “closed book” examination; no aids are permitted. No electronic or mechanical computing devices are permitted. Write your answers in the spaces provided. If necessary, use the backs of the pages for rough work. Show all steps and present all results clearly. State any assumptions that you may make. For full credit, answers to counting problems must be expressed as integers. Please write clearly; if we cannot read an answer, we cannot mark it. This examination has 12 pages (including this one). Be sure that you have a complete paper. Name Student Number MARKS Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Total Value 7 4 6 4 5 5 5 6 4 8 4 4 11 6 5 3 5 3 95 Mark Enter the frst letter oF your Family name here. 1
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1. (7 marks) Let A = { 1 , 2 , 3 } , let B = { 1 , { 2 }} , and let C = {∅} . (a) Find each set. i. A B { 1 } ii. A - B { 2 , 3 } iii. P ( B ) {∅ , { 1 } , {{ 2 }} , { 1 , { 2 }}} iv. B × C { (1 , ) , ( { 2 } , ) } (b) Indicate whether each statement is true or false by circling the appropriate word. Each correct answer is worth one mark while each incorrect answer will result in the loss of one mark. i. True or False 2 B ii. or False ∅⊆ C iii. or False ∅⊆P ( B ) 2. (4 marks) Let p , q , and r be statements. Show that p ( q r ) is logically equivalent to q ( p r ). pqr pq r p ( q r ) p rq ( p r ) TTT FT T T T TTF FF T T T TFT T T T TFF T T T FTT TT T T T FTF TF F F F FFT T T T FFF T F T Since the truth values of the two expressions are identical, they are logically equivalent. 2
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3. (6 marks) Let the domain of discourse be dogs in a park and let the predicates B ( x ), W ( x ), and L ( x,y )be : B ( x )=“ x barks loudly” W ( x x likes to wrestle” L ( x likes y (a) Express each of the statements below in simple English. i. x yL ( ) At least one dog likes all dogs. ii. x yB ( x ) W ( x ) L ( ) All dogs who bark loudly and like to wrestle like at least one dog. iii. xB ( x ) ∧∼ W ( x ) At least one dog barks loudly but doesn’t like to wrestle. (b) Write the following statements using predicates, quantiFers, and logical connectives. If nega- tions are used, express the statement so that no negation symbol is to the left of a quantiFer. i. Not all the dogs in the park bark loudly. x B ( x ) ii. At least one of the dogs in the park who barks loudly likes to wrestle. xB ( x ) W ( x ) iii. None of the dogs that Max (a dog) likes, bark loudly. xL (Max ,x ) →∼ B ( x ) 3
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4. (4 marks) Prove that for n Z ,i f5 n + 4 is an odd integer, then n is an odd integer. Proof by contraposition (We show that if n is an even integer, then 5 n + 4 is an even integer) Suppose that n is an even integer Then k Z ,n =2 k Then 5 n + 4 = 5(2 k ) + 4 = 2(5 k + 2), an even integer, as required QED 5. (5 marks)
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This homework help was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto- Toronto.

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f2006Soln - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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