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Discrete Math Structures

by: Mireya Heidenreich

Discrete Math Structures CS 2233

Mireya Heidenreich
GPA 3.55


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This 6 page Class Notes was uploaded by Mireya Heidenreich on Thursday October 29, 2015. The Class Notes belongs to CS 2233 at University of Texas at San Antonio taught by Staff in Fall. Since its upload, it has received 734 views. For similar materials see /class/231398/cs-2233-university-of-texas-at-san-antonio in ComputerScienence at University of Texas at San Antonio.

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Date Created: 10/29/15
Sample questions do not go to the answers before you tried to solve them Name Date Use the following to answer questions 172 In the questions below determine Whether the proposition is TRUE or FALSE 1 If it is raining then it is raining A True B False 21f2l3then23 l A True B False Use the following to answer questions 3 4 In the questions below suppose that Qx is x 1 Zr where x is a real number Find the truth value of the statement 3 VxQx A True B False 4 EXQCX A True B False Use the following to answer questions 576 In the questions below 10033 means x 2y xy where x and y are integers Determine the truth value of the statement 5 Vv3V Pxvt A line B False r 6 3xVyPxy A True B False Use the following to answer questions 7410 In the questions below suppose Penv is a predicate and the universe for the variables x and y is 123 Suppose P13 P2l P22 P23 P23 P31 P32 are true and Pxy is false otherwise Determine whether the following statements are true 7 VxEIyPxy A True B False 00 3xVyPxy A True B False gt5 ni l tm A PtvvxD A True B False O Vy3xPxy gt POEM A True B False Use the following to answer questions 1 12 In the questions below mark each statement TRUE or FALSE Assume that the statement applies to all sets 11 A B CA B C A True B False 12 AnBu CAuBmt UC A True B False Page 2 Use the following to answer questions 1344 In the questions below suppose A m39 and B vr Mark the stutcmcm TRUE 0r FALSE 13 x g B A True 13 False 14 Q E PB A True B False Use the following to answer questions 1517 In the questions below suppose A aim Mark the statement TRUE 0r FALSE Q5 g A X A A True B l ulse U 6 ur e A A True B False 17 1117 6 A Xl A True B False Use the following to answer questions 1820 In the quextiuns below Stlppox c A aim and B Aw Mark the statement TRUE or FA LS E 18 r e A B A True B False Page 3 J 2 l B Q A A True B False c g B A True B False Determine whether 7 gt 1 quotip gt q a qr Using 7 for it is cold and I for it is dryquot write quotIt is neither cold nor dryquot in symbols Use the following to answer questions 2324 In the questions below Pry means 3 and y are real numbers such that x 2y 2 5quot Determine whether the statement is true IQ Ut L O to 00 Q lt3 VxElyPxy 3xVyPry Consider the following theorem Ifx is an odd integer then 39 2 is odd Give a proof by contradiction of this theorem Consider the following theorem UN is an even integer then 21 l is odd Give a proof by contraposition of this theorem What is wrong with the following quotproofquot that 3 3 using backward reasoning Assume that 3 3 Squaring both sides yields 32 33 or 9 9 Therefore 3 3 Supposef N gt N has the ntle n 411 1 Determine whetherfis 1 1 Supposef N gt N has the rulen 4n 1 Determine whetherfis onto N J O Suppose Z A Z has the mle n 3nquot 1 Determine whetherfis 11 Page 4 3 b ix 3 LA 3 3 DJ 39J Supposcf Z gt Z has the rulen 3n ll Determine whetheris onto Z lquotind the sum 2 4 8 l6 1 32 233 Find the Sum 1 7 12 14 18 116 Find a proposition with lhrce variables p q and r that is true when cxuclly one ofilie three variables is true and false olhcrwisc 39 On Lhc island ofknights and knaves you encounter two pcopleA and 11 Person A says quot8 is a knavcquot Person 8 says quotWe are both knightsquot Determine whether each person is a knight or a knave Page 5 Answer Key 1 2 3 5 wwgtmgtgtgtwwmgtwgtgtamgtmgtgt Ix Ln Both truth tables are identical 2 gt q 11 gt 39 iC A 39 Tme since for every real number x we can find 21 real number 3 such that x 2y 5 namely y 5 x2 False if it were true for some number m then x0 5 239 for every y which is not possible V Suppose x is odd but 39 2 is even Therefore x 2k 1 and x 2 2 Hence 2k 1 2 21 Therefore 2k 1 I I even 2 odd a contradiction Suppose n l is even Therefore n 1 2k Therefore 2 2k I 2lt 1 1 which is odd The steps in the quotprooF39 cannot be reversed Knowing that the squares of two numbers A 3 and 3 are equal does not allow us to infer that the two numbers are equal Yer No No No Page 6


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