f2004Soln - 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 6, 2004 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 11 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 6 3 6 4 5 4 6 6 4 4 5 7 4 14 6 6 6 4 100 Mark Enter the frst letter oF your Family name here. 1
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1. (6 marks) Let A = { 1 , 2 , 3 , 4 , 5 } , B = { 3 , 6 } , and let E be the set of even integers. Find (a) A B = { 1 , 2 , 3 , 4 , 5 , 6 } (b) A - E = { 1 , 3 , 5 } (c) A E = { 2 , 4 } (d) P ( B )= {∅ , { 3 } , { 6 } , { 3 , 6 }} (e) B × {∅} = { (3 , ) , (6 , ) } (f) ( B A ) ( B E ) = False (or F) 2. (3 marks) Let p and q be statements. Use a truth table to show that ( p q ) ( q →∼ p )i sa tautology. pq p q q p q →∼ p ( p q ) ( q p ) TT TF F T T FT F F T T T T FF T T T 3. (6 marks) Let F ( x ) mean that student x is a frosh, and let C ( x, y ) mean that student x is enrolled in class y , where the domain of x is the set of all students at U of T and the domain of y is the set of all classes at U of T. Express the following statements in simple English. (a) C (Alice , ECE190) Alice is in ECE190. (b) x ( F ( x ) C ( x, FRS101)) All frosh are in FRS101. (c) x y z (( x ± = y ) ( C ( x, z ) C ( y,z ))) There are two students enrolled in the same set of classes. Write the following statements using predicates and quanti±ers, the negation symbol , and the logical connectives , , , and . If negations are used, express the statements so that no negation symbol is to the left of a quanti±er. (d) Bob is a frosh, but he is not enrolled in CIV101. F (Bob) ∧∼ C (Bob,CIV101) (e) Being a frosh is a necessary condition for any U of T student to be enrolled in ECE190. x ( C ( x, ECE190) F ( x )) (f) It is not the case that every student is enrolled in at least one course. x y C ( x, y ) 2
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4. (4 marks) Prove by contradiction that if the integers 1 , 2 , 3 ,..., 10 are separated into three groups, in any possible way, there exists at least one group whose sum is greater than or equal to 19.
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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.

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f2004Soln - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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