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f2005Soln

# f2005Soln - 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 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 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 4 8 7 6 4 4 96 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 = { 1 } (b) B - P = { 1 } (c) A P = { 3 } (d) P ( A - P )= {∅ , { 1 } , { 2 } , { 4 } , { 1 , 2 } , { 1 , 4 } , { 2 , 4 } , { 1 , 2 , 4 }} (e) A × {∅} = { (1 , ) , (2 , ) , (3 , ) , (4 , ) } (f) ( A - P ) B ( B - P ) A F T T 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. pq p q p q ( p q ) ( p q )( p q ) (( p q ) ( p q )) TT TF T T FT F T F T FF F T From the last column, we can see that the statement 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. i. x ( P ( x ) E ( x )) There is an even prime. ii. x y ( P ( x ) P ( y ) L ( ) →∼ E ( y )) The greater of any two diﬀerent primes is odd. iii. x y ( L ( ) P ( y )) Any positive integer is less than some prime. (b) Write the following statements using predicates, quantiFers, and logical connectives. If nega- tions are used, express the statement so that no negation symbol is to the left of a quantiFer. i. Not all positive integers are prime. ∼∀ xP ( x ) ≡∃ x P ( x ) ii. The product of two primes is not prime. x y zP ( x ) P ( y ) M ( ) →∼ P ( z ) iii. Every positive integer has a prime factor. x y zM ( y,z,x ) ( P ( y ) P ( z )) 3

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4. (5 marks) Consider the logical operator ± defned by the truth table below. x y x ± y T T F T F F F T F F F T Show how to represent each o± the expressions x , x y , and x y using no symbols other than the ±ollowing: xy ± () Consider the ±ollowing truth table: x ± y ( x ± y ) ± ( x ± y ) x ± ± y ( x ± x ) ± ( y ± y ) TT FTF FT TF FTT FF TFT From the truth table, we can see that x x ± xx y ( x ± y ) ± ( x ± y ) x y ( x ± x ) ± ( y ± y ) Alternatively, we can use the results ±or x and x y to get an expression ±or x y : x y ≡∼ ( x ∨∼ y ) ≡∼ (( x ± x ) ( y ± y )) ≡∼ ((( x ± x ) ± ( y ± y )) ± (( x ± x ) ± ( y ± y ))) ((( x
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f2005Soln - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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