TestBank
QuestionsforChapter1
Whatisthenegationofthepropositionsin1–4?
1. Abbyhasmorethan300friendsonFacebook.
2. AlissaownsmorequiltsthanFederico.
3. Amessagingpackageforacellphonecostslessthan$20permonth.
4. 4 5+2 5=6
Inquestions5–9,determinewhetherthepropositionisTRUEorFALSE.
5. 1+1=3ifandonlyif2+2=3.
6. Ifitisraining,thenitisraining.
7. If1 < 0,then3=4.
8. If2+1=3,then2=3 1.
9. If1+1=2or1+1=3,then2+2=3and2+2=4.
10. Writethetruthtablefortheproposition ¬(r →¬q) ∨ (p ∧¬r).
11.
(a) Findapropositionwiththetruthtableattheright.
(b) Findapropositionusingonly p,q, ¬ ,andtheconnective ∨ thathasthis truthtable.
12. Findapropositionwiththreevariables p , q ,and r thatistruewhen p and r aretrueand q isfalse,and falseotherwise.
13. Findapropositionwiththreevariables p , q ,and r thatistruewhenatmostoneofthethreevariablesis true,andfalseotherwise.
14. Findapropositionwiththreevariables p , q ,and r thatisnevertrue.
15. Findapropositionusingonly p,q, ¬ ,andtheconnective ∨ withthetruthtable attheright. p ¬p ?
In16–17,usetheconditional-disjunctionequivalencetofindanequivalentcompoundpropositionthatdoesnot involveconditions.
16. ¬p → q
17. p → (p ∧ q)
18. Determinewhether p → (q → r)and p → (q ∧ r)areequivalent.
19. Determinewhether p → (q → r)isequivalentto(p → q) → r
20. Determinewhether(p → q) ∧ (¬p → q) ≡ q .
21. Writeapropositionequivalentto p ∨¬q thatusesonly p,q, ¬ ,andtheconnective ∧
22. Writeapropositionequivalentto ¬p ∧¬q usingonly p,q, ¬ ,andtheconnective ∨
23. Provethattheproposition“ifitisnothot,thenitishot”isequivalentto“itishot.”
24. Writeapropositionequivalentto p → q usingonly p,q, ¬ ,andtheconnective ∨ .
25. Writeapropositionequivalentto p → q usingonly p,q, ¬ ,andtheconnective ∧
26. Provethat p → q anditsconversearenotlogicallyequivalent.
27. Provethat ¬p →¬q anditsinversearenotlogicallyequivalent.
Determinewhetherthefollowingtwopropositionsarelogicallyequivalent:
Determinewhetherthefollowingtwopropositionsarelogicallyequivalent:
30. Provethat(q ∧ (p →¬q)) →¬p isatautologyusingpropositionalequivalenceandthelawsoflogic.
31. Determinewhetherthispropositionisatautology:((p → q) ∧¬p) →¬q
32. Determinewhetherthispropositionisatautology:((p →¬q) ∧ q) →¬p .
In33–39,writethestatementintheform“If ... ,then ... .”
33. x isevenonlyif y isodd.
34. A implies B
35. Itishotwheneveritissunny.
36. Togetagoodgradeitisnecessarythatyoustudy.
37. Studyingissufficientforpassing.
38. Theteamwinsifthequarterbackcanpass.
39. Youneedtoberegisteredinordertocheckoutlibrarybooks.
40. Writethecontrapositive,converse,andinverseofthefollowing:Ifyoutryhard,thenyouwillwin.
41. Writethecontrapositive,converse,andinverseofthefollowing:YousleeplateifitisSaturday.
In42–44writethenegationofthestatement.(Don’twrite“Itisnottruethat .”)
42. ItisThursdayanditiscold.
43. Iwillgototheplayorreadabook,butnotboth.
44. Ifitisrainy,thenwegotothemovies.
45. Explainwhythenegationof“AlandBillareabsent”isnot“AlandBillarepresent.”
46. Using c for“itiscold”and d for“itisdry,”write“Itisneithercoldnordry”insymbols.
47. Using c for“itiscold”and r for“itisrainy,”write“Itisrainyifitisnotcold”insymbols.
48. Using c for“itiscold”and w for“itiswindy,”write“Tobewindyitisnecessarythatitbecold”insymbols.
49. Using c for“itiscold,” r for“itisrainy,”and w for“itiswindy,”write“Itisrainyonlyifitiswindyand cold”insymbols.
50. Express r ⊕ d inEnglish,where r is“itisrainy”and d is“itisdry.”
51. Translatethegivenstatementintopropositionallogicusingthepropositionsprovided:Oncertainhighways intheWashington,DCmetroareayouareallowedtotravelonhighoccupancylanesduringrushhouronly ifthereareatleastthreepassengersinthevehicle.Expressyouranswerintermsof r:“Youaretraveling duringrushhour.” t:“Youareridinginacarwithatleastthreepassengers.”and h:“Youcantravelona highoccupancylane.”
52. Asetofpropositionsis consistent ifthereisanassignmentoftruthvaluestoeachofthevariablesinthe propositionsthatmakeseachpropositiontrue.Isthefollowingsetofpropositionsconsistent?
Thesystemisinmultiuserstateifandonlyifitisoperatingnormally. Ifthesystemisoperatingnormally,thekernelisfunctioning. Thekernelisnotfunctioningorthesystemisininterruptmode. Ifthesystemisnotinmultiuserstate,thenitisininterruptmode. Thesystemisininterruptmode.
53. WhatBooleansearchcouldyouusetolookforwebpagesaboutU.S.nationalforestsnotinAlaskaorHawaii?
54. Ontheislandofknightsandknavesyouencountertwopeople, A and B .Person A says“ B isaknave.”
Person B says“Wearebothknights.”Determinewhethereachpersonisaknightoraknave.
55. Ontheislandofknightsandknavesyouencountertwopeople, A and B .Person A says“ B isaknave.”
Person B says“Atleastoneofusisaknight.”Determinewhethereachpersonisaknightoraknave. Questions56–58relatetoinhabitantsofanislandonwhichtherearethreekindsofpeople:knightswhoalways tellthetruth,knaveswhoalwayslie,andspieswhocaneithertellthetruthorlie.Youencounterthreepeople, A , B ,and C .Youknowoneofthethreepeopleisaknight,oneisaknave,andoneisaspy.Eachofthethreepeople knowsthetypeofpersoneachoftheothertwois.Foreachofthesesituations,ifpossible,determinewhetherthere isauniquesolution,listallpossiblesolutionsorstatethattherearenosolutions.
56. A says“Iamnotaknight,” B says“Iamnotaspy,”and C says“Iamnotaknave.”
57. A says“Iamaspy,” B says“Iamaspy”and C says“ B isaspy.”
58. A says“Iamaknight,” B says“Iamaknave,”and C says“Iamnotaknave.”
Findtheoutputofthecombinatorialcircuitsin59–60.
59. p q r 60. p q q r
Constructacombinatorialcircuitusinginverters,ORgates,andANDgates,thatproducestheoutputsin61–62 frominputbits p,q and r .
61. (¬p ∧¬q) ∨ (p ∧¬r)
62. ((p ∨¬q) ∧ r) ∧ ((¬p ∧¬q) ∨ r)
Determinewhetherthecompoundpropositionsin63–64aresatisfiable.
63. (¬p ∨¬q) ∧ (p → q)
64. (p → q) ∧ (q →¬p) ∧ (p ∨ q)
In65–67supposethat Q(x)is“ x +1=2x ,”where x isarealnumber.Findthetruthvalueofthestatement.
65. Q(2)
66. ∀xQ(x)
67. ∃xQ(x)
In68–75 P (x,y)means“ x +2y = xy ,”where x and y areintegers.Determinethetruthvalueofthestatement.
68. P (1, 1)
69. P (0, 0)
70. ∃yP (3,y)
71. ∀x∃yP (x,y)
72. ∃x∀yP (x,y)
73. ∀y∃xP (x,y)
74. ∃y∀xP (x,y)
75. ¬∀x∃y ¬P (x,y)
In76–77,expressthenegationofthestatementintermsofquantifierswithoutusingthenegationsymbol.
76. ∀x((x> 1) ∨ (x< 1))
77. ∃x(3 <x ≤ 7)
In78–79 P (x,y)means“ x and y arerealnumberssuchthat x +2y =5.”Determinewhetherthestatementis true.
78. ∀x∃yP (x,y)
79. ∃x∀yP (x,y)
In80–82 P (m,n)means“ m ≤ n ,”wheretheuniverseofdiscoursefor m and n isthesetofnonnegativeintegers. Whatisthetruthvalueofthestatement?
80. ∀nP (0,n)
81. ∃n∀mP (m,n)
82. ∀m∃nP (m,n)
Inquestions83–88suppose P (x,y)isapredicateandtheuniverseforthevariables x and y is {1, 2, 3} .Suppose P (1, 3), P (2, 1), P (2, 2), P (2, 3), P (3, 1), P (3, 2)aretrue,and P (x,y)isfalseotherwise.Determinewhether thefollowingstatementsaretrue.
83. ∀x∃yP (x,y)
84. ∃x∀yP (x,y)
85. ¬∃x∃y (P (x,y) ∧¬P (y,x))
86. ∀y∃x (P (x,y) → P (y,x))
87. ∀x∀y (x = y → (P (x,y) ∨ P (y,x))
88. ∀y∃x (x ≤ y ∧ P (x,y))
In88–92supposethevariable x representsstudentsand y representscourses,and:
U (y): y isanupper-levelcourse M (y): y isamathcourse F (x): x isafreshman
B(x): x isafull-timestudent T (x,y):student x istakingcourse y Writethestatementusingthesepredicatesandanyneededquantifiers.
89. EricistakingMTH281.
90. Allstudentsarefreshmen.
91. Everyfreshmanisafull-timestudent.
92. Nomathcourseisupper-level.
In93–95supposethevariable x representsstudentsand y representscourses,and:
U (y): y isanupper-levelcourse M (y): y isamathcourse F (x): x isafreshman
A(x): x isapart-timestudent T (x,y):student x istakingcourse y Writethestatementusingthesepredicatesandanyneededquantifiers.
93. Everystudentistakingatleastonecourse.
94. Thereisapart-timestudentwhoisnottakinganymathcourse.
95. Everypart-timefreshmanistakingsomeupper-levelcourse.
In96–98supposethevariable x representsstudentsand y representscourses,and: F (x): x isafreshman A(x): x isapart-timestudent T (x,y): x istaking y .
WritethestatementingoodEnglishwithoutusingvariablesinyouranswers.
96. F (Mikko)
97. ¬∃yT (Joe,y)
98. ∃x (A(x) ∧¬F (x))
In99–101supposethevariable x representsstudentsand y representscourses,and: M (y): y isamathcourse F (x): x isafreshman B(x): x isafull-timestudent T (x,y): x istaking y WritethestatementingoodEnglishwithoutusingvariablesinyouranswers.
99. ∀x∃yT (x,y)
100. ∃x∀yT (x,y)
101. ∀x∃y [(B(x) ∧ F (x)) → (M (y) ∧ T (x,y))]
In102–104supposethevariables x and y representrealnumbers,and L(x,y): x<yG(x): x> 0 P (x): x isaprimenumber. WritethestatementingoodEnglishwithoutusinganyvariablesinyouranswer.
102. L(7, 3)
103. ∀x∃yL(x,y)
104. ∀x∃y [G(x) → (P (y) ∧ L(x,y))]
In105–107supposethevariables x and y representrealnumbers,and L(x,y): x<yQ(x,y): x = yE(x): x iseven I(x): x isaninteger. Writethestatementusingthesepredicatesandanyneededquantifiers.
105. Everyintegeriseven.
106. If x<y ,then x isnotequalto y
107. Thereisnolargestrealnumber.
In108–109supposethevariables x and y representrealnumbers,and E(x): x iseven G(x): x> 0 I(x): x isaninteger.
Writethestatementusingthesepredicatesandanyneededquantifiers.
108. Somerealnumbersarenotpositive.
109. Noevenintegersareodd.
In110–112supposethevariable x representspeople,and F (x): x isfriendly T (x): x istall A(x): x isangry. Writethestatementusingthesepredicatesandanyneededquantifiers.
110. Somepeoplearenotangry.
111. Alltallpeoplearefriendly.
112. Nofriendlypeopleareangry.
In113–114supposethevariable x representspeople,and F (x): x isfriendly T (x): x istall A(x): x isangry.
Writethestatementusingthesepredicatesandanyneededquantifiers.
113. Sometallangrypeoplearefriendly.
114. Ifapersonisfriendly,thenthatpersonisnotangry.
In115–117supposethevariable x representspeople,and F (x): x isfriendly T (x): x istall A(x): x isangry. WritethestatementingoodEnglish.Donotusevariablesinyouranswer.
115. A(Bill)
116. ¬∃x (A(x) ∧ T (x))
117. ¬∀x (F (x) → A(x))
In118–120supposethevariable x representsstudentsandthevariable y representscourses,and A(y): y isanadvancedcourse S(x): x isasophomore F (x): x isafreshman T (x,y): x istaking y . Writethestatementusingthesepredicatesandanyneededquantifiers.
118. Thereisacoursethateveryfreshmanistaking.
119. Nofreshmanisasophomore.
120. Somefreshmanistakinganadvancedcourse.
In121–122supposethevariable x representsstudentsandthevariable y representscourses,and A(y): y isanadvancedcourse F (x): x isafreshman T (x,y): x istaking yP (x,y): x passed y Writethestatementusingtheabovepredicatesandanyneededquantifiers.
121. Nooneistakingeveryadvancedcourse.
122. Everyfreshmanpassedcalculus.
In123–125supposethevariable x representsstudentsandthevariable y representscourses,and T (x,y): x istaking yP (x,y): x passed y WritethestatementingoodEnglish.Donotusevariablesinyouranswers.
123. ¬P (Wisteria, MAT100)
124. ∃y∀xT (x,y)
125. ∀x∃yT (x,y)
In126–130assumethattheuniversefor x isallpeopleandtheuniversefor y isthesetofallmovies.Writethe Englishstatementusingthefollowingpredicatesandanyneededquantifiers:
S(x,y): x saw yL(x,y): x liked yA(y): y wonanaward C(y): y isacomedy.
126. Nocomedywonanaward.
127. Loissaw Casablanca,butdidn’tlikeit.
128. Somepeoplehaveseeneverycomedy.
129. Noonelikedeverymoviehehasseen.
130. Benhasneverseenamoviethatwonanaward.
In131–133assumethattheuniversefor x isallpeopleandtheuniversefor y isthesetofallmovies.Writethe statementingoodEnglish,usingthepredicates S(x,y): x saw yL(x,y): x liked y Donotusevariablesinyouranswer.
131. ∃y ¬S(Margaret,y)
132. ∃y∀xL(x,y)
133. ∀x∃yL(x,y)
In134–143supposethevariable x representsstudents, y representscourses,and T (x,y)means“ x istaking y .” MatchtheEnglishstatementwithallitsequivalentsymbolicstatementsinthislist:
1. ∃x∀yT (x,y)2. ∃y∀xT (x,y)3. ∀x∃yT (x,y)
4. ¬∃x∃yT
134. Everycourseisbeingtakenbyatleastonestudent.
135. Somestudentistakingeverycourse.
136. Nostudentistakingallcourses.
137. Thereisacoursethatallstudentsaretaking.
138. Everystudentistakingatleastonecourse.
139. Thereisacoursethatnostudentsaretaking.
140. Somestudentsaretakingnocourses.
141. Nocourseisbeingtakenbyallstudents.
142. Somecoursesarebeingtakenbynostudents.
143. Nostudentistakinganycourse.
In144–154supposethevariable x representsstudents, F (x)means“ x isafreshman,”and M (x)means“ x isa mathmajor.”MatchthestatementinsymbolswithoneoftheEnglishstatementsinthislist:
1. Somefreshmenaremathmajors.
2. Everymathmajorisafreshman.
3. Nomathmajorisafreshman.
144. ∀x (M (x) →¬F (x))
145. ¬∃x (M (x) ∧¬F (x))
146. ∀x (F (x) →¬M (x))
147. ∀x (M (x) → F (x))
148. ∃x (F (x) ∧ M (x))
149. ¬∀x (¬F (x) ∨¬M (x))
150. ∀x (¬(M (x) ∧¬F (x)))
151. ∀x (¬M (x) ∨¬F (x))
152. ¬∃x (M (x) ∧¬F (x))
153. ¬∃x (M (x) ∧ F (x))
154. ¬∀x (F (x) →¬M (x))
In155–158let F (A)bethepredicate“ A isafiniteset”and S(A,B)bethepredicate“ A iscontainedin B .” Supposetheuniverseofdiscourseconsistsofallsets.Translatethestatementintosymbols.
155. Notallsetsarefinite.
156. Everysubsetofafinitesetisfinite.
157. Noinfinitesetiscontainedinafiniteset.
158. Theemptysetisasubsetofeveryfiniteset.
In158–163writethenegationofthestatementingoodEnglish.Don’twrite“Itisnottruethat .... ”
159. Somebananasareyellow.
160. Allintegersendinginthedigit7areodd.
161. Notestsareeasy.
162. Rosesareredandvioletsareblue.
163. SomeskiersdonotspeakSwedish.
164. Astudentisaskedtogivethenegationof“allbananasareripe.”
(a) Thestudentresponds“allbananasarenotripe.”ExplainwhytheEnglishinthestudent’sresponseis ambiguous.
(b) Anotherstudentsaysthatthenegationofthestatementis“nobananasareripe.”Explainwhythisis notcorrect.
(c) Anotherstudentsaysthatthenegationofthestatementis“somebananasareripe.”Explainwhythis isnotcorrect.
(d) Givethecorrectnegation.
165. Explainwhythenegationof“Somestudentsinmyclassusee-mail”isnot“Somestudentsinmyclassdo notusee-mail.”
166. Whatistheruleofinferenceusedinthefollowing: Ifitsnowstoday,theuniversitywillbeclosed.Theuniversitywillnotbeclosedtoday.Therefore,itdidnot snowtoday.
167. Whatistheruleofinferenceusedinthefollowing: IfIworkallnightonthishomework,thenIcanansweralltheexercises.IfIansweralltheexercises,I willunderstandthematerial.Therefore,ifIworkallnightonthishomework,thenIwillunderstandthe material.
168. Explainwhyanargumentofthefollowingformisnotvalid:
169. Determinewhetherthefollowingargumentisvalid:
170. Determinewhetherthefollowingargumentisvalid:
171. Showthatthehypotheses“IleftmynotesinthelibraryorIfinishedtheroughdraftofthepaper”and“I didnotleavemynotesinthelibraryorIrevisedthebibliography”implythat“Ifinishedtheroughdraftof thepaperorIrevisedthebibliography.”
172. Determinewhetherthefollowingargumentisvalid.Nametheruleofinferenceorthefallacy.
If n isarealnumbersuchthat n> 1,then n2 > 1.Supposethat n2 > 1.Then n> 1.
173. Determinewhetherthefollowingargumentisvalid.Nametheruleofinferenceorthefallacy.
If n isarealnumbersuchthat n> 2,then n2 > 4.Supposethat n ≤ 2.Then n2 ≤ 4.
174. Determinewhetherthefollowingargumentisvalid: SheisaMathMajororaComputerScienceMajor. Ifshedoesnotknowdiscretemath,sheisnotaMathMajor. Ifsheknowsdiscretemath,sheissmart. SheisnotaComputerScienceMajor. Therefore,sheissmart.
175. Determinewhetherthefollowingargumentisvalid. Rainydaysmakegardensgrow. Gardensdon’tgrowifitisnothot. Italwaysrainsonadaythatisnothot. Therefore,ifitisnothot,thenitishot.
176. Determinewhetherthefollowingargumentisvalid. Ifyouarenotinthetennistournament,youwillnotmeetEd.
Ifyouaren’tinthetennistournamentorifyouaren’tintheplay,youwon’tmeetKelly. YoumeetKellyoryoudon’tmeetEd.
Itisfalsethatyouareinthetennistournamentandintheplay. Therefore,youareinthetennistournament.
177. Showthatthepremises“Everystudentinthisclasspassedthefirstexam”and“Alvinaisastudentinthis class”implytheconclusion“Alvinapassedthefirstexam.”
178. Showthatthepremises“Jeanisastudentinmyclass”and“NostudentinmyclassisfromEngland”imply theconclusion“JeanisnotfromEngland.”
179. Determinewhetherthepremises“Somemathmajorsleftthecampusfortheweekend”and“Allseniorsleft thecampusfortheweekend”implytheconclusion“Someseniorsaremathmajors.”
180. Showthatthepremises“Everyonewhoreadthetextbookpassedtheexam,”and“Edreadthetextbook” implytheconclusion“Edpassedtheexam.”
181. Determinewhetherthepremises“Nojuniorsleftcampusfortheweekend”and“Somemathmajorsarenot juniors”implytheconclusion“Somemathmajorsleftcampusfortheweekend.”
182. Showthatthepremise“MydaughtervisitedEuropelastweek”impliestheconclusion“Someonevisited Europelastweek.”
183. Supposeyouwishtoproveatheoremoftheform“if p then q .”
(a) Ifyougiveadirectproof,whatdoyouassumeandwhatdoyouprove?
(b) Ifyougiveaproofbycontraposition,whatdoyouassumeandwhatdoyouprove?
(c) Ifyougiveaproofbycontradiction,whatdoyouassumeandwhatdoyouprove?
184. Supposethatyouhadtoproveatheoremoftheform“if p then q .”Explainthedifferencebetweenadirect proofandaproofbycontraposition.
185. Giveadirectproofofthefollowing:“If x isanoddintegerand y isaneveninteger,then x + y isodd.”
186. Giveaproofbycontradictionofthefollowing:“If n isanoddinteger,then n2 isodd.”
187. Considerthefollowingtheorem:“if x and y areoddintegers,then x + y iseven.”Giveadirectproofof thistheorem.
188. Considerthefollowingtheorem:“if x and y areoddintegers,then x + y iseven.”Giveaproofby contradictionofthistheorem.
189. Giveaproofbycontradictionofthefollowing:If x and y areevenintegers,then xy iseven.
190. Considerthefollowingtheorem:If x isanoddinteger,then x +2isodd.Giveadirectproofofthistheorem
191. Considerthefollowingtheorem:If x isanoddinteger,then x +2isodd.Giveaproofbycontrapositionof thistheorem.
192. Considerthefollowingtheorem:If x isanoddinteger,then x +2isodd.Giveaproofbycontradictionof thistheorem.
193. Considerthefollowingtheorem:If n isaneveninteger,then n +1isodd.Giveadirectproofofthis theorem.
194. Considerthefollowingtheorem:If n isaneveninteger,then n +1isodd.Giveaproofbycontraposition ofthistheorem.
195. Considerthefollowingtheorem:If n isaneveninteger,then n +1isodd.Giveaproofbycontradictionof thistheorem.
196. Provethatthefollowingistrueforallpositiveintegers n : n isevenifandonlyif3n2 +8iseven.
197. Provethefollowingtheorem: n isevenifandonlyif n2 iseven.
198. Prove:if m and n areevenintegers,then mn isamultipleof4.
199. Proveordisprove:Forallrealnumbers x and y , x y = x − y
200. Proveordisprove:Forallrealnumbers x and y , x + x = 2x
201. Proveordisprove:Forallrealnumbers x and y , xy = x · y .
202. Giveaproofbycasesthat x ≤|x| forallrealnumbers x
203. Useaproofbycasestoshowthat27isnotthesquareofapositiveinteger.
204. Supposeyouareallowedtogiveeitheradirectprooforaproofbycontrapositionofthefollowing:if3n +5 iseven,then n isodd.Whichtypeofproofwouldbeeasiertogive?Explainwhy.
205. Provethatthefollowingthreestatementsaboutpositiveintegers n areequivalent: (a) n iseven; (b) n3 +1isodd; (c) n2 1isodd.
206. Givenany40people,provethatatleastfourofthemwereborninthesamemonthoftheyear.
207. Provethattheequation2x2 + y2 =14hasnopositiveintegersolutions.
208. Whatiswrongwiththefollowing“proof”that 3=3,usingbackwardreasoning?Assumethat 3=3. Squaringbothsidesyields( 3)2 =32 ,or9=9.Therefore 3=3.
AnswersforChapter1
1. Abbyhasfewerthan301friendsonfacebook.
2. AllisadoesnotownmorequiltsthanFederico.
3. Amessagingpackageforacellphonecostsatleast$20permonth.
4. 4.5+2.5 =6
5. True
6. True
p) ∨ (q ∧¬q) ∨ (r ∧¬r)
15. ¬(¬p ∨ q) ∨¬(p ∨¬q)
16. p ∨ q
17. ¬p ∨ q
18. Notequivalent.Let q befalseand p and r betrue.
19. Notequivalent.Let p , q ,and r befalse.
20. Bothtruthtablesareidentical:
pq (p → q) ∧ (¬p → q) q
21. ¬(¬p ∧ q)
22. ¬(p ∨ q)
23. Bothpropositionsaretruewhen“itishot”istrueandbotharefalsewhen“itishot”isfalse.
24. ¬p ∨ q
25. ¬(p ∧¬q)
26. Truthvaluesdifferwhen p istrueand q isfalse.
27. Truthvaluesdifferwhen p isfalseand q istrue.
28. No 29. Yes
31. No 32. Yes
33. If x iseven,then y isodd.
34. If A ,then B
35. Ifitissunny,thenitishot.
36. Ifyoudon’tstudy,thenyoudon’tgetagoodgrade(equivalently,ifyougetagoodgrade,thenyoustudy).
37. Ifyoustudy,thenyoupass.
38. Ifthequarterbackcanpass,thentheteamwins.
39. Ifyouarenotregistered,thenyoucannotcheckoutlibrarybooks(equivalently,ifyoucheckoutlibrary books,thenyouareregistered).
40. Contrapositive:Ifyouwillnotwin,thenyoudonottryhard.Converse:Ifyouwillwin,thenyoutryhard. Inverse:Ifyoudonottryhard,thenyouwillnotwin.
41. Contrapositive:Ifyoudonotsleeplate,thenitisnotSaturday.Converse:Ifyousleeplate,thenitis Saturday.Inverse:IfitisnotSaturday,thenyoudonotsleeplate.
42. ItisnotThursdayoritisnotcold.
43. Iwillgototheplayandreadabook,orIwillnotgototheplayandnotreadabook.
44. Itisrainyandwedonotgotothemovies.
45. Bothpropositionscanbefalseatthesametime.Forexample,AlcouldbepresentandBillabsent.
50. “Itisrainyoritisdry,butitcannotbeboth.”
51. (r ∧ t) → h
52. Using m , n , k ,and i ,therearethreerowsofthetruthtablethathaveallfivepropositionstrue:therows TTTT,FFTT,FFFTfor m,n,k,i .
53. U.S. AND NATIONAL AND FOREST AND (NOT ALASKA) AND (NOT HAWAII)
54. A isaknight, B isaknave.
55. A isaknave, B isaknight.
56. A isthespy, B istheknight,and C istheknave.
57. A istheknave, B isthespy,and C istheknight.
58. A istheknight, B isthespy,and C istheknave,or A istheknave, B isthespy,and C istheknight.
59. ¬(¬p ∨ q) ∧ r
60. ¬(p ∧¬q) ∧ (q ∨ r)
x wecanfindarealnumber y suchthat x +2y =5,namely y =(5 x)/2.
x0 ,then x0 =5 2y forevery y ,whichisnotpossible.
92. ∀y (M (y)→¬U (y))
93. ∀x∃yT (x,y)
94. ∃x∀y [A(x) ∧ (M (y) →¬T (x,y))]
95. ∀x∃y [(F (x) ∧ A(x)) → (U (y) ∧ T (x,y))]
96. Mikkoisafreshman.
97. Joeisnottakinganycourse.
98. Somepart-timestudentsarenotfreshmen.
99. Everystudentistakingacourse.
100. Somestudentistakingeverycourse.
101. Everyfull-timefreshmanistakingamathcourse.
102. 7 < 3.
103. Thereisnolargestnumber.
104. Nomatterwhatpositivenumberischosen,thereisalargerprime.
105. ∀x (I(x)→E(x))
106. ∀x∀y (L(x,y)→¬Q(x,y))
107. ∀x∃yL(x,y)
108. ∃x ¬G(x)
109. ¬∃x (I(x) ∧ E(x) ∧¬E(x)])
110. ∃x ¬A(x)
111. ∀x (T (x)→F (x))
112. ∀x (F (x)→¬A(x))
113. ∃x (T (x) ∧ A(x) ∧ F (x))
114. ∀x (F (x)→¬A(x))
115. Billisangry.
116. Nooneistallandangry.
117. Somefriendlypeoplearenotangry.
118. ∃y∀x (F (x)→T (x,y))
119. ¬∃x (F (x) ∧ S(x)]
120. ∃x∃y (F (x) ∧ A(y) ∧ T (x,y))
121. ¬∃x∀y (A(y)→T (x,y))
122. ∀x (F (x)→P (x,calculus))
123. WisteriadidnotpassMAT100.
124. Thereisacoursethatallstudentsaretaking.
125. Everystudentistakingatleastonecourse.
126. ∀y (C(y)→¬A(y))
127. S(Lois,Casablanca) ∧¬L(Lois,Casablanca)
128. ∃x∀y [C(y)→S(x,y)]
129. ¬∃x∀y [S(x,y)→L(x,y)]
130. ¬∃y [A(y) ∧ S(Ben,y)]
131. ThereisamoviethatMargaretdidnotsee.
132. Thereisamoviethateveryoneliked.
133. Everyonelikedatleastonemovie.
155. ∃A ¬F (A) 156. ∀A ∀B [(F (B) ∧ S(A,B))→F (A)] 157. ¬∃A ∃B (¬F (A) ∧ F (B) ∧ S(A,B))
158. ∀A (F (A)→S(∅,A))
159. Nobananasareyellow.
160. Someintegersendinginthedigit7arenotodd.
161. Sometestsareeasy.
162. Rosesarenotredorvioletsarenotblue.
163. AllskiersspeakSwedish.
164. (a) Dependingonwhichwordisemphasized,thesentencecanbeinterpretedas“allbananasarenon-ripe fruit”(i.e.,nobananasareripe)oras“notallbananasareripe”(i.e.,somebananasarenotripe).
(b) Bothstatementscanbefalseatthesametime.
(c) Bothstatementscanbetrueatthesametime.
(d) Somebananasarenotripe.
165. Bothstatementscanbetrueatthesametime.
166. Modustollens
167. Hypotheticalsyllogism
168. Setting p falseand q trueyieldtruehypothesesbutafalseconclusion.
169. Notvalid: p false, q false, r true
170. Notvalid: p true, q true, r true
171. Useresolutionon l ∨ f and ¬ l ∨ r toconclude f ∨ r
172. Notvalid:fallacyofaffirmingtheconclusion
173. Notvalid:fallacyofdenyingthehypothesis
174. Valid
175. Valid
176. Notvalid
177. Universalinstantiation
178. Universalinstantiation
179. Thetwopremisesdonotimplytheconclusion.
180. Let R(x)bethepredicate“ x hasreadthetextbook”and P (x)bethepredicate“ x passedtheexam.”The followingistheproof:
1. ∀x (R(x) → P (x))hypothesis
2. R(Ed) → P (Ed)universalinstantiationon1
3. R(Ed)hypothesis
4. P (Ed)modusponenson2and3
181. Thetwopremisesdonotimplytheconclusion.
182. Existentialgeneralization
183. (a) Assume p ,prove q
(b) Assume ¬q ,prove ¬p
(c) Assume p ∧¬q ,showthatthisleadstoacontradiction.
184. Directproof:Assume p ,show q .Indirectproof:Assume ¬q ,show ¬p .
185. Suppose x =2k +1, y =2l .Therefore x + y
+ l)+1,whichisodd. 186. Suppose n =2k +1but n2 =2l .Therefore(2k +1)2 =2l ,or4k2 +4k +1=2l .Hence2(2k2 +2k l)= 1 (even=odd),acontradiction.Therefore n2 isodd.
187. Let x =2k +1, y =2l +1.Therefore x + y =2k +1+2l +1=2(k + l +1),whichiseven.
188. Suppose x =2k +1and y =2l +1,but x + y =2m +1.Therefore(2k +1)+(2l +1)=2m +1.Hence 2(k + l m +1)=1(even=odd),whichisacontradiction.Therefore x + y iseven.
189. Suppose x =2k and y =2l ,but xy =2m +1.Therefore2k · 2l =2m +1.Hence2(2kl m)=1(even= odd),whichisacontradiction.Therefore xy iseven.
190. Let x =2k +1.Therefore x +2=2k +1+2=2(k +1)+1,whichisodd.
191. Suppose x +2=2k .Therefore x =2k 2=2(k 1),whichiseven.
192. Suppose x isoddbut x +2iseven.Therefore x =2k +1and x +2=2l .Hence(2k +1)+2=2l .Therefore 2(k +1 l)= 1(even=odd),acontradiction.
193. Let n =2k .Therefore n +1=2k +1,whichisodd.
194. Suppose n +1iseven.Therefore n +1=2k .Therefore n =2k 1=2(k 1)+1,whichisodd.
195. Suppose n =2k but n +1=2l .Therefore2k +1=2l (even=odd),whichisacontradiction.
196. If n iseven,then n =2k .Therefore3n2 +8=3(2k)2 +8=12k2 +8=2(6k2 +4),whichiseven.If n is odd,then n =2k +1.Therefore3n2 +8=3(2k +1)2 +8=12k2 +12k +11=2(6k2 +6k +5)+1,which isodd.
197. If n iseven,then n2 =(2k)2 =2(2k2),whichiseven.If n isodd,then n2 =(2k +1)2 =2(2k2 +2k)+1, whichisodd.
198. If m =2k and n =2l ,then mn =4kl .Hence mn isamultipleof4.
199. False: x =2, y =1/2
200. False: x =1/2
201. False: x =3/2, y =3/2
202. Case1, x ≥ 0:then x = |x| ,so x ≤|x| .Case2, x< 0:here x< 0and0 < |x| ,so x< |x|
203. Weprovethestatementbycases.Thetwocases1 ≤ x ≤ 5or x ≥ 6areexhaustive.Inthefirstcase,if 1 ≤ x ≤ 5,then x2 ≤ 25so x2 =27.Inthesecondcase,if x ≥ 6,then x2 ≥ 36and x2 =27.So x2 =27 forallpositiveintegers x andweconcludethat27isnotaperfectsquare.
204. Itiseasiertogiveacontrapositionproof;itisusuallyeasiertoproceedfromasimpleexpression(suchas n ) toamorecomplexexpression(suchas3n +5iseven).Beginbysupposingthat n isnotodd.Therefore n isevenandhence n =2k forsomeinteger k .Therefore3n +5=3(2k)+5=6k +5=2(3k +2)+1,which isnoteven.Ifwetryadirectproof,weassumethat3n +5iseven;thatis,3n +5=2k forsomeinteger k . Fromthisweobtain n =(2k 5)/3,andititnotobviousfromthisformthat n iseven.
205. Provethat(a)and(b)areequivalentandthat(a)and(c)areequivalent.
206. Ifatmostthreepeoplewerebornineachofthe12monthsoftheyear,therewouldbeatmost36people.
207. Giveaproofbycases.Thereareonlysixcasesthatneedtobeconsidered: x = y =1; x =1, y =2; x =1, y =3; x =2, y =1; x = y =2; x =2, y =3.
208. Thestepsinthe“proof”cannotbereversed.Knowingthatthesquaresoftwonumbers, 3and3,areequal doesnotallowustoinferthatthetwonumbersareequal.
QuestionsforChapter2
Foreachofthepairsofsetsin1–3determinewhetherthefirstisasubsetofthesecond,thesecondisasubsetof thefirst,orneitherisasubsetoftheother.
1. ThesetofpeoplewhowerebornintheU.S.,thesetofpeoplewhoareU.S.citizens.
2. Thesetofstudentsstudyingaprogramminglanguage,thesetofstudentsstudyingJava.
3. Thesetofanimalslivingintheocean,thesetoffish.
4. Proveordisprove: A (B ∩ C)=(A B) ∪ (A C).
5. Provethat A ∩ B = A ∪ B bygivingacontainmentproof(thatis,provethattheleftsideisasubsetof therightsideandthattherightsideisasubsetoftheleftside).
6. Provethat A ∩ B = A ∪ B bygivinganelementtableproof.
7. Provethat A ∩ B = A ∪ B bygivingaproofusinglogicalequivalence.
8. Provethat A ∩ B = A ∪ B bygivingaVenndiagramproof.
9. Provethat A ∩ (B ∪ C)=(A ∩ B) ∪ (A ∩ C)bygivingacontainmentproof(thatis,provethattheleftside isasubsetoftherightsideandthattherightsideisasubsetoftheleftside).
10. Provethat A ∩ (B ∪ C)=(A ∩ B) ∪ (A ∩ C)bygivinganelementtableproof.
11. Provethat A ∩ (B ∪ C)=(A ∩ B) ∪ (A ∩ C)bygivingaproofusinglogicalequivalence.
12. Provethat A ∩ (B ∪ C)=(A ∩ B) ∪ (A ∩ C)bygivingaVenndiagramproof.
13. Proveordisprove:if A , B ,and C aresets,then A (B ∩ C)=(A B) ∩ (A C).
14. Proveordisprove A ⊕ (B ⊕ C)=(A ⊕ B) ⊕ C
Inquestions15–18useaVenndiagramtodeterminewhichrelationship, ⊆ ,=,or ⊇ ,istrueforthepairofsets.
15. A ∪ B , A ∪ (B A)
16. A ∪ (B ∩ C),(A ∪ B) ∩ C
17. (A B) ∪ (A C), A (B ∩ C)
18. (A C) (B C), A B
Inquestions19–23determinewhetherthegivensetisthepowersetofsomeset.Ifthesetisapowerset,givethe setofwhichitisapowerset.
19. {∅
21. {∅, {a}, {∅,a}}
22. {∅, {a}, {∅}, {a, ∅}}
23. {∅, {a, ∅}}
24. Provethat S ∪ T = S ∩ T forallsets S and T
In25–35markeachstatementTRUEorFALSE.Assumethatthestatementappliestoallsets.
25. A (B C)=(A B) C
26. (A C) (B C)= A B
27. A ∪ (B ∩ C)=(A ∪ B) ∩ (A ∪ C)
28. A ∩ (B ∪ C)=(A ∪ B) ∩ (A ∪ C)
29. A ∪ B ∪ A = A
30. If A ∪ C = B ∪ C ,then A = B
31. If A ∩ C = B ∩ C ,then A = B
32. If A ∩ B = A ∪ B ,then A = B
33. If A ⊕ B = A ,then B = A
34. Thereisaset A suchthat |P(A)| =12.
35. A ⊕ A = A
36. Findthreesubsetsof {1, 2, 3, 4, 5, 6, 7, 8, 9} suchthattheintersectionofanytwohassize2andtheintersection ofallthreehassize1. 37. Find
41. Suppose U = {1, 2,..., 9}, A =allmultiplesof2, B =allmultiplesof3,and C = {3, 4, 5, 6, 7} .Find C (B A).
42. Suppose S = {1, 2, 3, 4, 5}.Find |P(S)|
Inquestions43–46suppose A = {x,y} and B = {x, {x}} .MarkthestatementTRUEorFALSE.
43. x ⊆ B
44. ∅∈P(B)
45. {x}⊆ A B
46. |P(A)| =4
Inquestions47–54suppose A = {a,b,c} .MarkthestatementTRUEorFALSE.
47. {b,c}∈P(A)
48. {{a}}⊆P(A)
49. ∅⊆ A
50. {∅}⊆P(A)
51. ∅⊆ A × A
52. {a,c}∈ A
53. {a,b}∈ A × A
54. (c,c) ∈ A × A
Inquestions55–62suppose A = {1, 2, 3, 4, 5} .MarkthestatementTRUEorFALSE.
55. {1}∈P(A)
56. {{3}}⊆P(A)
57. ∅⊆ A
58. {∅}⊆P(A)
59. ∅⊆P(A)
60. {2, 4}∈ A × A
61. {∅}∈P(A)
62. (1, 1) ∈ A × A
Inquestions63–66,supposethefollowingaremultisets:
63. Find S ∪ T .
64. Find S ∩ T 65. Find S T . 66. Find S + T
Inquestions67–69supposethefollowingarefuzzysets:
67. Find F and R 68. Find F ∪ R .
69. Find F ∩ R
Inquestions70–79,suppose A = {a,b,c} and B = {b, {c}} .MarkthestatementTRUEorFALSE.
70. c ∈ A B
71. |P(A × B)| =64
72. ∅∈P(B)
73. B ⊆ A
74. {c}⊆ B
75. {a,b}∈ A × A
76. {b,c}∈P(A)
77. {b, {c}}∈P(B)
78. ∅⊆ A × A
79. {{{c}}}⊆P(B)
80. Find A2 if A = {1,a}.
Inquestions81–93determinewhetherthesetisfiniteorinfinite.Ifthesetisfinite,finditssize.
81. { x | x ∈ Z and x2 < 10 }
82. P({a,b,c,d}),where P denotesthepowerset
83. {1, 3, 5, 7,...}
84. A × B ,where A = {1, 2, 3, 4, 5} and B = {1, 2, 3}
85. { x | x ∈ N and9x2 1=0 }
86. P(A),where A isthepowersetof {a,b,c}
87. A × B ,where A = {a,b,c} and B = ∅
88. { x | x ∈ N and4x2 8=0 }
89. { x | x ∈ Z and x2 =2 }
90. P(A),where A = P({1, 2})
91. {1, 10, 100, 1000,...}
92. S × T ,where S = {a,b,c} and T = {1, 2, 3, 4, 5}
93. { x | x ∈ Z and x2 < 8 }
94. Provethatbetweeneverytworationalnumbers a/b and c/d (a) thereisarationalnumber. (b) thereareaninfinitenumberofrationalnumbers.
95. Provethatthereisnosmallestpositiverationalnumber.
96. ConsiderthesefunctionsfromthesetoflicenseddriversinthestateofNewYork.Isafunctionone-to-one ifitassignstoalicenseddriverhisorher
(a) birthdate (b) mother’sfirstname (c) driverslicensenumber?
In97–98determinewhethereachofthefollowingsetsiscountableoruncountable.Forthosethatarecountably infiniteexhibitaone-to-onecorrespondencebetweenthesetofpositiveintegersandthatset.
97. Thesetofpositiverationalnumbersthatcanbewrittenwithdenominatorslessthan3.
98. Thesetofirrationalnumbersbetween √2and π/2.
99. AdapttheCantordiagonalizationargumenttoshowthatthesetofpositiverealnumberslessthan1with decimalrepresentationsconsistingonlyof0sand1sisuncountable.
100. Showthat(0, 1)hasthesamecardinalityas(0, 2).
101. Showthat(0, 1]and R havethesamecardinality.
Inquestions102–110determinewhethertheruledescribesafunctionwiththegivendomainandcodomain.
g(n)=anyinteger >n
F : R → R,where F (x)= 1 x 5
117. Suppose f : N → N hastherule f (n)=4n +1.Determinewhether f is1-1.
118. Suppose f : N → N hastherule f (n)=4n +1.Determinewhether f isonto N
119. Suppose f : Z → Z hastherule f (n)=3n2
f is1-1.
120. Suppose f : Z → Z hastherule f (n)=3n 1.Determinewhether f isonto Z
121. Suppose f : N → N hastherule f (n)=3n2 1.Determinewhether f is1-1.
122. Suppose f : N → N hastherule f (n)=4n2 +1.Determinewhether f isonto N
123. Suppose f : R → R where f (x)= x/2 .
(a) Drawthegraphof f (b) Is f 1-1? (c) Is f onto R?
124. Suppose f : R → R where f (x)= x/2
(a) If S = { x | 1 ≤ x ≤ 6 } ,find f (S). (b) If T = {3, 4, 5} ,find f 1(T ).
125. Determinewhether f isafunctionfromthesetofallbitstringstothesetofintegersif f (S)istheposition ofa1bitinthebitstring S
126. Determinewhether f isafunctionfromthesetofallbitstringstothesetofintegersif f (S)isthenumber of0bitsin S
127. Determinewhether f isafunctionfromthesetofallbitstringstothesetofintegersif f (S)isthelargest integer i suchthatthe i thbitof S is0and f (S)=1when S istheemptystring(thestringwithnobits).
128. Let f (x)= x 3/3 .Find f (S)if S is:
129. Suppose f : R → Z where f (x)= 2x 1 .
(a) Drawthegraphof f (b) Is f 1-1?(Explain) (c) Is f onto Z?(Explain)
130. Suppose f : R → Z where f (x)= 2x 1
(a) If A = {x | 1 ≤ x ≤ 4},find f (A).
(c) If C = {−9, 8} ,find f 1(C). (d) If
131. Suppose g : R → R where g(x)= x 1 2
(a) Drawthegraphof g (b) Is g 1-1? (c) Is g onto R?
132. Suppose g : R → R where g(x)= x 1 2 .
(a) If S = {x | 1 ≤ x ≤ 6} ,find g(S). (b) If T = {2} ,find g 1(T ).
133. Showthat x = − −x .
134. Proveordisprove:Forallpositiverealnumbers x and y , x y ≤ x · y .
135. Proveordisprove:Forallpositiverealnumbers x and y , x · y ≤ x · y
136. Suppose g : A → B and f : B → C where A = {1, 2, 3, 4} , B = {a,b,c}, C = {2, 7, 10},and f and g are definedby g = {(1,b), (2,a), (3,a), (4,b)} and f = {(a, 10), (b, 7), (c, 2)}.Find f ◦ g .
137. Suppose g : A → B and f : B → C where A = {1, 2, 3, 4} , B = {a,b,c}, C = {2, 7, 10},and f and g are definedby g = {(1,b), (2,a), (3,a), (4,b)} and f = {(a, 10), (b, 7), (c, 2)}.Find f 1
Inquestions138–141,supposethat g : A → B and f : B → C ,where A = B = C = {1, 2, 3, 4} , g = {(1, 4), (2, 1), (3, 1), (4, 2)} ,and f = {(1,
138. Find f ◦ g .
139. Find g ◦ f .
140. Find g ◦
141.