f2006Solns

# f2006Solns - 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 25th, 2006 Examiners: John Carter, Periklis Papakonstantinou 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 14 15 Total Value 10 5 4 7 6 4 3 6 2 7 8 3 4 7 6 82 Mark Enter the frst letter oF your Family name here. 1

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1. (a) Let A = { 1 , 2 } and let B = {{ 2 } , 3 } . 10 marks i. Find A B ii. Find A - B { 1 , 2 } iii. Find P ( B ) {∅ , {{ 2 }} , { 3 } , {{ 2 } , 3 }} iv. True or false? {∅} ∈ P ( A ) False v. True or false? {∅} ⊆ P ( A ) True (b) If A = { a,b,c } , B = { a,c } , and C = { b } , give the values of each of the following sets: i. A × B { ( a,a ) , ( ) , ( b,a ) , ( b,c ) , ( c,a ) , ( c,c ) } ii. | A × B × C | 6 iii. A × ( B × C ) { ( a, ( a,b )) , ( a, ( c, b )) , ( b, ( , ( b, ( c,b , ( c, ( )) , ( c, ( )) } iv. A ×∅ (c) Let N be the set of non-empty subsets of some set S . Express this de±nition of N using set notation. N = { X | X S X ± = ∅} or N = P ( S ) - {∅} 2
2. Let p , q , and r be the following statements. 5 marks p = “It is sunny” q = “It is cold” r = “It is pleasant to walk” 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 sunny but not warm. p q (b) It is warm only if it is sunny. q p (c) If it is warm or sunny then it is pleasant to walk.

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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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f2006Solns - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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