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f2005Exam

# f2005Exam - 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 7, 2005 Examiners: John Carter, Paul McCabe 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 13 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 19 Total Value 6 4 6 5 4 5 5 4 6 6 3 5 4 3 8 7 6 4 4 95 Mark Enter the frst letter oF your Family name here. 1

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1. (6 marks) Let A = { 1 , 2 , 3 , 4 } , let B = { 1 , 3 , 5 , 7 } , and let P be the set of odd prime numbers. Find (a) A B (b) B - P (c) A P (d) P ( A - P ) (e) A × {∅} (f) ( A - P ) B ( B - P ) A 2. (4 marks) Let p and q be statements. Use a truth table to determine whether or not ( p q ) (( p ∨∼ q ) ( p q )) is a tautology. 2
3. (6 marks) In the domain of positive integers, let the predicates P ( x ), L ( x,y ), E ( x ), and M ( x,y,z )be : P ( x )=“ x is prime” L ( x is less than y E ( x x is even” M ( x × y = z (a) Express each of the statements below in simple English.

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