F2005Solns - UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F Discrete Mathematics Midterm Test October 24th 2005 Examiners

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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F — Discrete Mathematics Midterm Test October 24th, 2005 Examiners: John Carter, Paul McCabe SOLUTIONS This is a “closed book” test; 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 test 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 Total Value 5 4 8 4 5 6 3 4 6 8 10 10 7 6 86 Mark Enter the frst letter oF your Family name here. 1
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1. (5 marks) Let p , q , and r be the following statements. p = “It is winter” q = “There is snow on the ground” r = “It is cold outside” Write each of the following statements using only the symbols p , q , and r , the negation symbol , and the logical connectives , , , , and . (a) It is winter, but there is no snow on the ground. p ∧∼ q (b) There is snow on the ground only if it is winter. q p (c) If there is snow on the ground, then it is cold outside. q r (d) Either there is no snow on the ground or it is not cold outside or it is winter. q ∨∼ r p (e) If it is not cold outside, then either there is no snow on the ground or it is not winter. r ( q p ) 2. (4 marks) Let p , q , and r be statements. Show that ( p q ) ( q r ) is logically equivalent to p q r . pqrp qq r ( p q ) ( q r ) p q r TTT F T T T TTF F T T T TFT T T T T TFF T F F F FTT T T T T FTF T T T T FFT F T T T FFF F F T T Alternatively, ( p q ) ( q r ) ≡∼ (( p q ) ( p q )) ( q r ) (( p q ) ( p q )) ( q r ) (( p q q ) ( p q q )) r (( p q ) ( p
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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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F2005Solns - UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F Discrete Mathematics Midterm Test October 24th 2005 Examiners

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