f2004Solns - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F — Discrete Mathematics Midterm Test October 28th, 2004 Examiners: John Carter, Ben Liang 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 10 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 Total Value 6 4 10 5 5 3 4 8 8 10 10 7 4 84 Mark Enter the frst letter oF your Family name here. 1
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1. (6 marks) Let p , q , and r be the following statements: p : The grass is green. q : The maple leaves are red. r : It is autumn. Write these statements using p , q , and r , the negation symbol , and the logical connectives , , , and . (a) The grass is green but the maple leaves are red. p q (b) Either the grass is green or the maples leaves are red, but not both. p q (c) If it is autumn, then the maple leaves are red. r q (d) If the grass is not green, then it is autumn. p r (e) That it is autumn is a necessary condition for maple leaves to be red. q r (f) If the grass is not green and the maple leaves are red, then it is autumn. p q r 2. (4 marks) Let p , q , and r be statements. Show that ( p q ) r is logically equivalent to p ( q r ). Solution #1, using a truth table: p q r p q ( p q ) r q r p ( q r ) T T T T T T T T T F T F F F T F T F T T T T F F F T T T F T T F T T T F T F F T F T F F T F T T T F F F F T T T The 5th and 7th columns have the same truth values. Solution #2: ( p q ) r ≡∼ ( p q ) r ( p ∨∼ q ) r ≡∼ p ( q r ) ≡∼ p ( q r ) p ( q r ) 2
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3. (10 marks) Let the domain of the variables x and y be the birds in a forest. Let F ( x )be“ x can Fy,” let S ( x x can sing,” and let R ( x, y x is a friend of y .” Suppose that Tweedy is a bird in the forest. Use quanti±ers and logical connectives to express each of these statements.
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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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f2004Solns - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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