Discrete Mathematics with Applications, 5th Edition by
Susanna S. Epp
Chapter1
Test Bank Questions
1.Fillintheblankstorewritethefollowingstatementwithvariables:Isthereanintegerwitha remainderof1whenitisdividedby4andaremainderof3whenitisdividedby7?
(a)Isthereaninteger n suchthat n has ?
(b)Doesthereexist suchthatif n isdividedby4theremainderis1andif ?
2.Fill intheblankstorewritethefollowingstatementwithvariables: Givenanypositiverealnumber,thereisapositiverealnumberthatissmaller.
(a)Givenanypositiverealnumber r,thereis s suchthat s is .
(b)Forany , suchthat s<r
3.Rewritethefollowingstatementlessformally,withoutusingvariables: Thereisaninteger n suchthat1/n isalsoaninteger.
4.Fillintheblankstorewritethefollowingstatement: Forallobjects T ,if T isatrianglethen T hasthreesides.
(a)Alltriangles
(b)Everytriangle
(c)If anobjectisatriangle,thenit
(d)If T , then T
(e)Foralltriangles T ,
5.Fill intheblankstorewritethefollowingstatement: Everyrealnumberhasanadditiveinverse.
(a)Allrealnumbers
(b)Foranyrealnumber x,thereis for x.
(c)Forallrealnumbers x,thereisrealnumber y suchthat
6.Fill intheblankstorewritethefollowingstatement: Thereisapositiveintegerthatislessthanorequaltoeverypositiveinteger.
(a)Thereisapositiveinteger m suchthat m is . (b)There isa suchthat everypositiveinteger.
(c)Thereisapositiveinteger m whichsatisfiesthepropertythatgivenanypositiveinteger n, m is .
7.(a) Writeinwordshowtoreadthefollowingoutloud {n ∈ Z | n isafactorof9} (b)Usetheset-rosternotationtoindicatetheelementsintheset.
8. (a)Is {5}∈{1, 3, 5}?
(b)Is {5}⊆{1, 3, 5}?
(c)Is {5}∈{{1}, {3}, {5}}?
(d)Is {5}⊆{{1}, {3}, {5}}?
9. Let A = {a,b,c} and B = {u,v}.Write a A × B and b B × A.
10. Let A = {3, 5, 7} and B = {15, 16, 17, 18},anddefinearelation R from A to B asfollows:For all(x,y) ∈ A × B, (x,y) ∈ R ⇔ y x isaninteger
(a) Is3 R 15?Is3 R 16?Is(7, 17) ∈ R?Is(3, 18) ∈ R?
(b) Write R asasetoforderedpairs.
(c) Writethedomainandco-domainof R
(d) Drawanarrowdiagramfor R
(e) Is R afunctionfrom A to B?Explain.
11. Definearelation R from R to R asfollows:Forall(x,y) ∈ R × R,(x,y) ∈ R if,andonlyif, x = y2 +1
(a) Is(2, 5) ∈ R?Is(5, 2) ∈ R?Is( 3) R 10?Is10 R ( 3)?
(b) Drawthegraphof R intheCartesianplane.
(c) Is R afunctionfrom R to R?Explain.
12. Let A = {1, 2, 3, 4} and B = {a,b,c} Defineafunction G: A → B asfollows: G = {(1,b), (2,c), (3,b), (4,c)}
(a) Find G(2).
(b) Drawanarrowdiagramfor G
13. Definefunctions F and G from R to R bythefollowingformulas:
F (x)=(x +1)(x 3)and G(x)=(x 2)2 7
Does F = G?Explain.
Chapter2
1. Whichofthefollowingisanegationfor“JimisinsideandJanisatthepool.”
(a) JimisinsideorJanisnotatthepool.
(b) JimisinsideorJanisatthepool.
(c) JimisnotinsideorJanisatthepool.
(d) JimisnotinsideandJanisnotatthepool.
(e) JimisnotinsideorJanisnotatthepool.
2. Whichofthefollowingisanegationfor“JimhasgrownorJoanhasshrunk.”
(a) JimhasgrownorJoanhasshrunk.
(b) JimhasgrownorJoanhasnotshrunk.
(c) JimhasnotgrownorJoanhasnotshrunk.
(d) JimhasgrownandJoanhasshrunk.
(e) JimhasnotgrownandJoanhasnotshrunk.
(f) JimhasgrownandJoanhasnotshrunk.
3. Writeanegationforeachofthefollowingstatements:
(a) Thevariable S isundeclaredandthedataareoutoforder.
(b) Thevariable S isundeclaredorthedataareoutoforder.
(c) IfAlwaswithBobonthefirst,thenAlisinnocent.
(d) 5 ≤ x< 2(where x isaparticularrealnumber)
4. Arethefollowingstatementformslogicallyequivalent: p ∨ q → p and p ∨ (∼ p ∧ q)?Includea truthtableandafewwordsexplaininghowthetruthtablesupportsyouranswer.
5. Stateprecisely(butconcisely)whatitmeansfortwostatementformstobelogicallyequivalent.
6. Writethefollowingtwostatementsinsymbolicformanddeterminewhethertheyarelogically equivalent.Includeatruthtableandafewwordsexplaininghowthetruthtablesupports youranswer.
IfSamboughtitatCrownBooks,thenSamdidn’tpayfullprice. SamboughtitatCrownBooksorSampaidfullprice.
7. Writethefollowingtwostatementsinsymbolicformanddeterminewhethertheyarelogically equivalent.Includeatruthtableandafewwordsexplaininghowthetruthtablesupports youranswer.
IfSamisoutofSchlitz,thenSamisoutofbeer.
SamisnotoutofbeerorSamisnotoutofSchlitz.
8. Writetheconverse,inverse,andcontrapositiveof“IfAnnisJan’smother,thenJoseisJan’s cousin.”
9. Writetheconverse,inverse,andcontrapositiveof“IfEdisSue’sfather,thenLiuisSue’s cousin.”
10. Writetheconverse,inverse,andcontrapositiveof“IfAlisTom’scousin,thenJimisTom’s grandfather.”
11. Rewritethefollowingstatementinif-thenformwithoutusingtheword“necessary”:Getting ananswerof10forproblem16isanecessaryconditionforsolvingproblem16correctly.
12. Stateprecisely(butconcisely)whatitmeansforaformofargumenttobevalid.
13. Considertheargumentform:
Usethetruthtablebelowtodeterminewhetherthisformofargumentisvalidorinvalid. Annotatethetable(asappropriate)andincludeafewwordsexplaininghowthetruthtable supportsyouranswer.
14. Considertheargumentform:
Usethetruthtablebelowtodeterminewhetherthisargumentformisvalidorinvalid.Annotate thetable(asappropriate)andincludeafewwordsexplaininghowthetruthtablesupports youranswer.
15. Determinewhetherthefollowingargumentisvalidorinvalid.Includeatruthtableandafew wordsexplainingwhythetruthtableshowsvalidityorinvalidity.
IfHugoisaphysicsmajororifHugoisamathmajor,thenheneedstotakecalculus. HugoneedstotakecalculusorHugoisamathmajor. Therefore,HugoisaphysicsmajororHugoisamathmajor.
16. Determinewhetherthefollowingargumentisvalidorinvalid.Includeatruthtableandafew wordsexplainingwhythetruthtableshowsvalidityorinvalidity.
If12divides709,438then3divides709,438. Ifthesumofthedigitsof709,438isdivisibleby9then3divides709,438. Thesumofthedigitsof709,438isnotdivisibleby9. Therefore,12doesnotdivide709,438.
17. Writetheformofthefollowingargument.Istheargumentvalidorinvalid?Justifyyour answer.
If54,587isaprimenumber,then17isnotadivisorof54,587. 17isadivisorof54,587. Therefore,54,587isnotaprimenumber.
18. Writetheformofthefollowingargument.Istheargumentvalidorinvalid?Justifyyour answer.
IfAnnhastheflu,thenAnnhasafever. Annhasafever. Therefore,Annhastheflu.
19. Ontheislandofknightsandknaves,youmeetthreenatives,A,B,andC,whoaddressyou asfollows:
A:Atleastoneofusisaknave.
B:Atmosttwoofusareknaves. WhatareA,B,andC?
20. Considerthefollowingcircuit.
(a) Findtheoutputofthecircuitcorrespondingtotheinput P =1,Q =0, and R =1 (b) WritetheBooleanexpressioncorrespondingtothecircuit.
21. Write1101012 indecimalform.
22. Write75inbinarynotation.
23. DrawthecircuitthatcorrespondstothefollowingBooleanexpression:(P ∧ Q) ∨ (∼ P ∧ ∼ Q).(Noteforstudentswhohavestudiedsomecircuitdesign:Donotsimplifythecircuit;just drawtheonethatexactlycorrespondstotheexpression.)
24. Findacircuitwiththefollowinginput/outputtable.
25. Find101112 +10112
26. Write1001102 indecimalform.
27. Writethe8-bittwo’scomplementfor49.
Discrete Mathematics with Applications, 5th Edition
by Susanna S. Epp
AnswersforTestBankQuestions:Chapters1-4
Pleaseusecautionwhenusingtheseanswers.Smalldifferencesinwording,notation,orchoiceofexamples orcounterexamplesmaybeacceptable.
Chapter1
1.a.aremainderof1whenitisdividedby4andaremainderof3whenitisdividedby7
b.aninteger n; n isdividedby7theremainderis3
2.a.apositiverealnumber;smallerthan r
b.positiverealnumber r;thereisapositiverealnumber s
Fillintheblankstorewritethefollowingstatementwithvariables:
3.Thereisanintegerwhosereciprocalisalsoaninteger.
4.a.havethreesides
b.hasthreesides
c.hasthreesides
d.isatriangle;hasthreesides
e. T hasthreesides
5.a.haveadditiveinverses
b.anadditiveinverse
c. y isanadditiveinversefor x
6.a.lessthanorequaltoeverypositiveinteger
b.positiveinteger m;lessthanorequaltoeverypositiveinteger
c.lessthanorequalto n
7.(a)Thesetofallintegers n suchthat n isafactorof9.
Or: Thesetofallelements n in Z suchthat n isafactorof9.
Or: Thesetofallelements n inthesetofallintegerssuchthat n isafactorof9.
(b) {1, 3, 9}
8.(a)No (b)Yes (c)Yes (d)No
9.a. {(a,u), (a,v), (b,u), (b,v), (c,u), (c,v)}
b. {(u,a), (v,a), (u,b), (v,b), (u,c), (v,c)}
10.a.Yes;No;No;Yes
b. {(3, 15), (3, 18), (5, 15)}
c.domainis {3, 5, 7};co-domainis {15, 16, 17, 18}
d.Drawanarrowdiagramfor R
e.No: R failsbothconditionsforbeingafunctionfrom A to B.(1)Elements5and7in A arenot relatedtoanyelementsin B,and(2)thereisanelementin A,namely3,thatisrelatedtotwodifferent elementsin B,namely15and18.
11. a.No;Yes;No;Yes
b.Drawthegraphof R intheCartesianplane.
c.No: R failsbothconditionsforbeingafunctionfrom R to R.(1)Therearemanyelementsin R thatarenotrelatedtoanyelementin R.Forinstance,noneof0,1/2,and 1isrelatedtoanyelement of R.(2)thereareelementsin R thatarerelatedtotwodifferentelementsin R.Forinstance2is relatedtoboth1and 1.
12. a. G(2)= c
b.Drawanarrowdiagramfor G
13. F = G. Notethatforeveryrealnumber x,
whereas
Thus,forinstance,
Chapter2
1. e
2. e
3. a.Thevariable S isnotundeclaredorthedataarenotoutoforder.
b.Thevariable S isnotundeclaredandthedataarenotoutoforder. c.AlwaswithBobonthefirst,andAlisnotinnocent.
d. 5 >x or x ≥ 2
4. Thestatementformsarenotlogicallyequivalent.
Truthtable:
Explanation:Thetruthtableshowsthat p ∨ q → p and p ∨ (∼ p ∧ q)havedifferenttruthvaluesin rows3and4,i.e,when p isfalse.Therefore p ∨ q → p and p ∨ (∼ p ∧ q)arenotlogicallyequivalent.
5. Sampleanswers:
Twostatementformsarelogicallyequivalentif,andonlyif,theyalwayshavethesametruthvalues. Or: Twostatementformsarelogicallyequivalentif,andonlyif,nomatterwhatstatementsare substitutedinaconsistentwayfortheirstatementvariablestheresultingstatementshavethesame truthvalue.
6. Solution1: Thegivenstatementsarenotlogicallyequivalent.Let p be“SamboughtitatCrown Books,”and q be“Samdidn’tpayfullprice.”Thenthetwostatementshavethefollowingform: p → q and p∨∼ q.
Thetruthtablesforthesestatementformsare