FINAL EXAM STUDY GUIDE QUESTIONS
WITH COMPLETE ANSWERS
True mmor mmfalse? mm"Mozart mmis mma mmgreat mmcomposer" mmis mma mmproposition. mm-
mmAnswerfasle mm- mm0pinion
What mmis mmthe mmcorrect mmmeaning mmof mmthe mmlogical mmexpression mmp→q∨r∧s mm?
a. mm((p→q)∨r)∧s mm
b. mm(p→(q∨r))∧s mm
c. mmp→(q∨(r∧s)) mm
d. mmp→((q∨r)∧s) mm- mmAnswerc
Consider mmthe mmtwo mmstatements:
1. mm(p mm→ mmq) mm∨ mm(p mm→ mm¬q)
2. mm(p mm→ mmq) mm∨ mm(¬p mm→ mmq).
a. mmNeither mmis mma mmtautology. mm
b. mm2. mmis mma mmtautology, mm1. mmis mmnot. mm
c. mmThey mmare mmboth mmtautologies. mm
d. mm1. mmis mma mmtautology, mm2. mmis mmnot. mm- mmAnswerc
1. mmis mma mmtautology mmbecause mmq mmtrue mmmakes mm(p mm→ mmq) mmtrue mmand mmq
mmfalse mmmakes mm(p mm→ mm¬q) mmtrue, mmregardless mmof mmthe mmtruth mmvalue mmof
mmp. mmThus mm(p mm→ mmq) mm∨ mm(p mm→ mm¬q) mmis mmalways mmtrue.
,2. mmis mma mmtautology mmbecause mmp mmfalse mmmakes mm(p mm→ mmq) mmtrue, mmand
mmp mmtrue mmmakes mm(¬p mm→ mmq) mmtrue, mmregardless mmof mmthe mmtruth mmvalue
mmof mmq. mmThus mm(p mm→ mmq) mm∨ mm(¬p mm→ mmq) mmis mmalways mmtrue.
Consider mmthe mmtwo mmstatements
1. mm(p mm→ mmq) mm∧ mm(p mm→ mm¬q)
2. mm(p mm→ mmq) mm∧ mm(¬p mm→ mmq).
a. mmneither mmis mma mmcontradiction. mm
b. mm2. mmis mma mmcontradiction, mm1. mmis mmnot. mm
c. mmThey mmare mmboth mmcontradictions. mm
d. mm1. mmis mma mmcontradiction, mm2. mmis mmnot. mm- mmAnswera
You mmcan mmsee mmthat mm1 mmis mmnot mma mmcontradiction mmbecause mmp mmfalse
mmmakes mmit mmtrue.
You mmcan mmsee mmthat mm2 mmis mmnot mma mmcontradiction mmbecause mmq mmtrue
mmmakes mmit mmtrue.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Converse: mm- mmAnswerConverse: mmIf mmits mma mmrectangle, mmthen mmit's mma
mmsquare.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Inverse: mm- mmAnswerIf mmits mmnot mma mmsquare, mmthen mmits mmnot mma mmrectangle.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Contrapositive: mm- mmAnswerIf mmit's mma mmnot mma mmrectangle, mmthen mmits mmnot mma
mmsquare.
True mmor mmfalse? mmThe mmtwo mmstatements
,(p mm∧ mmq) mm→ mmr
and
(p mm→ mmr) mm∨ mm(q mm→ mmr)
are mmlogically mmequivalent. mm- mmAnswerTrue
Select mmwhich mmone mmof mmthe mmfollowing mmstatements mmis mmlogically mmequivalent
mmto mm(p mm→ mmq) mm→ mmr:
a. mm(p mm∧ mmq) mm∨ mm¬r mm
mm
b. mm(p mm∧ mmq) mm∨ mmr mm
c. mm(p mm∧ mm¬q) mm∨ mmr mm
mm
d. mm(¬p mm∧ mmq) mm∨ mmr mm- mmAnswerc
Given mmreal mmnumbers mmx mmand mmy, mmfind mmthe mminverse mmof
x mm> mm0 mm∧ mmy mm≥ mm0 mm→ mmx mm+ mmy mm> mm0.
a. mmx mm< mm0 mm∧ mmy mm≤ mm0 mm→ mmx mm+ mmy mm< mm0 mm
mm
b. mm¬x mm> mm0 mm∧ mm¬y mm≥ mm0 mm→ mm¬x mm+ mm¬y mm> mm0 mm
c. mmx mm≤ mm0 mm∨ mmy mm< mm0 mm→ mmx mm+ mmy mm≤ mm0 mm
mm
d. mmx mm≤ mm0 mm∧ mmy mm< mm0 mm→ mmx mm+ mmy mm≤ mm0 mm
mm
e. mmx mm< mm0 mm∨ mmy mm≤ mm0 mm→ mmx mm+ mmy mm< mm0 mm- mmAnswerc
What mmis mmthe mmnegation mmof mm"Chris mmhas mma mmcat mmor mma mmdog"?
a. mmChris mmhas mmneither mmcat mmnor mmdog. mm
b. mmChris mmdoesn't mmhave mma mmpet. mm
c. mmChris mmdoesn't mmhave mma mmcat, mmor mmChris mmdoesn't mmhave mma mmdog. mm
d. mmChris mmhas mma mmcat mmbut mmnot mma mmdog, mmor mmChris mmhas mma mmdog mmbut
mmnot mma mmcat. mm- mmAnswera
True mmor mmfalse? mm"Having mmmoney mmis mma mmnecessary mmcondition mmfor
mmhappiness" mmmeans mmthat mmhaving mmmoney mmguarantees mmhappiness. mm-
mmAnswerfalse
, "Having mmmoney mmis mma mmnecessary mmcondition mmfor mmhappiness" mmdoes mmnot
mmmean mmthat mmhaving mmmoney mmguarantees mmhappiness. mmIt mmmeans mmthat
mmhaving mmmoney mmis mma mmprecondition mmfor mmhappiness. mmIt mmis mmsaying mmthat
mmif mmyou mmare mmhappy, mmthen mmyou mmmust mmhave mmmoney.
True mmor mmfalse? mm"Absence mmof mmevidence mmof mmillness mmis mmnecessary mmfor
mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor" mmmeans mmthat mmif mmyou
mmhave mmno mmevidence mmof mmillness, mma mmdoctor mmwill mmgive mmyou mma mmclean
mmbill mmof mmhealth. mm- mmAnswerfalse
The mmstatement mmis mmfalse. mm"Absence mmof mmevidence mmof mmillness mmis
mmnecessary mmfor mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor" mmonly
mmmeans mmthat mmlack mmof mmevidence mmof mmillness mmis mma mmprecondition mmfor
mmreceiving mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor, mmbut mmmay mmnot
mmguarantee mmit.
Select mmwhich mmof mmthe mmfollowing mmexplains mmthe mmconditional mmcorrectly.
mmThere mmmay mmbe mmmore mmthan mmone mmcorrect mmchoice. mmFor mmfull mmcredit
mmon mmthis mmquestion, mmyou mmmust mmcheck mmall mmcorrect mmexplanations mmand
mmnone mmof mmthe mmincorrect mmones.
(Precisely mmwhen mmmeans mmif mmand mmonly mmif. mmDisjunctions mmare mminclusive.)
a. mmThe mmconditional mmis mmfalse mmprecisely mmwhen mmthe mmpremise mmis mmtrue
mmand mmthe mmconclusion mmis mmfalse.
b. mmThe mmconditional mmis mmtrue mmprecisely mmwhen mmthe mmpremise mmis mmfalse
mmor mmthe mmconclusion mmis mmtrue.
c. mmThe mmconditional mmis mmfalse mmprecisely mmwhen mmthe mmpremise mmis mmfalse
mmand mmthe mmconclusion mmis mmtrue.
d. mmThe mmconditional mmis mmtrue mmprecisely mmwhen mmthe mmpremise mmis mmtrue mmor
mmthe mmconclusion mmis mmfalse. mm- mmAnswera mmand mmb
The mmconditional mmis mmbest mmunderstood mmas mma mmpromise: mmif mmA mmhappens,
mmthen mmB mmwill mmhappen. mmThe mmonly mmway mma mmpromise mmcan mmbe
mmconsidered mmbroken mmis mmif mmA mmhappened mmbut mmB mmdid mmnot mmhappen. mmIn
mmall mmother mmcases, mmwe mmconsider mmthe mmpromise mmkept, mmincluding mmthe
mmcase mmwhere mmA mmdid mmnot mmhappen. mmThen, mmthe mmpromise mmwas mmnever
mmput mmto mmthe mmtest, mmand mmis mmconsidered mmkept mmby mmdefault.
WITH COMPLETE ANSWERS
True mmor mmfalse? mm"Mozart mmis mma mmgreat mmcomposer" mmis mma mmproposition. mm-
mmAnswerfasle mm- mm0pinion
What mmis mmthe mmcorrect mmmeaning mmof mmthe mmlogical mmexpression mmp→q∨r∧s mm?
a. mm((p→q)∨r)∧s mm
b. mm(p→(q∨r))∧s mm
c. mmp→(q∨(r∧s)) mm
d. mmp→((q∨r)∧s) mm- mmAnswerc
Consider mmthe mmtwo mmstatements:
1. mm(p mm→ mmq) mm∨ mm(p mm→ mm¬q)
2. mm(p mm→ mmq) mm∨ mm(¬p mm→ mmq).
a. mmNeither mmis mma mmtautology. mm
b. mm2. mmis mma mmtautology, mm1. mmis mmnot. mm
c. mmThey mmare mmboth mmtautologies. mm
d. mm1. mmis mma mmtautology, mm2. mmis mmnot. mm- mmAnswerc
1. mmis mma mmtautology mmbecause mmq mmtrue mmmakes mm(p mm→ mmq) mmtrue mmand mmq
mmfalse mmmakes mm(p mm→ mm¬q) mmtrue, mmregardless mmof mmthe mmtruth mmvalue mmof
mmp. mmThus mm(p mm→ mmq) mm∨ mm(p mm→ mm¬q) mmis mmalways mmtrue.
,2. mmis mma mmtautology mmbecause mmp mmfalse mmmakes mm(p mm→ mmq) mmtrue, mmand
mmp mmtrue mmmakes mm(¬p mm→ mmq) mmtrue, mmregardless mmof mmthe mmtruth mmvalue
mmof mmq. mmThus mm(p mm→ mmq) mm∨ mm(¬p mm→ mmq) mmis mmalways mmtrue.
Consider mmthe mmtwo mmstatements
1. mm(p mm→ mmq) mm∧ mm(p mm→ mm¬q)
2. mm(p mm→ mmq) mm∧ mm(¬p mm→ mmq).
a. mmneither mmis mma mmcontradiction. mm
b. mm2. mmis mma mmcontradiction, mm1. mmis mmnot. mm
c. mmThey mmare mmboth mmcontradictions. mm
d. mm1. mmis mma mmcontradiction, mm2. mmis mmnot. mm- mmAnswera
You mmcan mmsee mmthat mm1 mmis mmnot mma mmcontradiction mmbecause mmp mmfalse
mmmakes mmit mmtrue.
You mmcan mmsee mmthat mm2 mmis mmnot mma mmcontradiction mmbecause mmq mmtrue
mmmakes mmit mmtrue.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Converse: mm- mmAnswerConverse: mmIf mmits mma mmrectangle, mmthen mmit's mma
mmsquare.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Inverse: mm- mmAnswerIf mmits mmnot mma mmsquare, mmthen mmits mmnot mma mmrectangle.
Identify mmthe mmconverse, mminverse mmand mmcontrapositive mmof:
If mmit's mma mmsquare, mmthen mmit mmis mma mmrectangle.
Contrapositive: mm- mmAnswerIf mmit's mma mmnot mma mmrectangle, mmthen mmits mmnot mma
mmsquare.
True mmor mmfalse? mmThe mmtwo mmstatements
,(p mm∧ mmq) mm→ mmr
and
(p mm→ mmr) mm∨ mm(q mm→ mmr)
are mmlogically mmequivalent. mm- mmAnswerTrue
Select mmwhich mmone mmof mmthe mmfollowing mmstatements mmis mmlogically mmequivalent
mmto mm(p mm→ mmq) mm→ mmr:
a. mm(p mm∧ mmq) mm∨ mm¬r mm
mm
b. mm(p mm∧ mmq) mm∨ mmr mm
c. mm(p mm∧ mm¬q) mm∨ mmr mm
mm
d. mm(¬p mm∧ mmq) mm∨ mmr mm- mmAnswerc
Given mmreal mmnumbers mmx mmand mmy, mmfind mmthe mminverse mmof
x mm> mm0 mm∧ mmy mm≥ mm0 mm→ mmx mm+ mmy mm> mm0.
a. mmx mm< mm0 mm∧ mmy mm≤ mm0 mm→ mmx mm+ mmy mm< mm0 mm
mm
b. mm¬x mm> mm0 mm∧ mm¬y mm≥ mm0 mm→ mm¬x mm+ mm¬y mm> mm0 mm
c. mmx mm≤ mm0 mm∨ mmy mm< mm0 mm→ mmx mm+ mmy mm≤ mm0 mm
mm
d. mmx mm≤ mm0 mm∧ mmy mm< mm0 mm→ mmx mm+ mmy mm≤ mm0 mm
mm
e. mmx mm< mm0 mm∨ mmy mm≤ mm0 mm→ mmx mm+ mmy mm< mm0 mm- mmAnswerc
What mmis mmthe mmnegation mmof mm"Chris mmhas mma mmcat mmor mma mmdog"?
a. mmChris mmhas mmneither mmcat mmnor mmdog. mm
b. mmChris mmdoesn't mmhave mma mmpet. mm
c. mmChris mmdoesn't mmhave mma mmcat, mmor mmChris mmdoesn't mmhave mma mmdog. mm
d. mmChris mmhas mma mmcat mmbut mmnot mma mmdog, mmor mmChris mmhas mma mmdog mmbut
mmnot mma mmcat. mm- mmAnswera
True mmor mmfalse? mm"Having mmmoney mmis mma mmnecessary mmcondition mmfor
mmhappiness" mmmeans mmthat mmhaving mmmoney mmguarantees mmhappiness. mm-
mmAnswerfalse
, "Having mmmoney mmis mma mmnecessary mmcondition mmfor mmhappiness" mmdoes mmnot
mmmean mmthat mmhaving mmmoney mmguarantees mmhappiness. mmIt mmmeans mmthat
mmhaving mmmoney mmis mma mmprecondition mmfor mmhappiness. mmIt mmis mmsaying mmthat
mmif mmyou mmare mmhappy, mmthen mmyou mmmust mmhave mmmoney.
True mmor mmfalse? mm"Absence mmof mmevidence mmof mmillness mmis mmnecessary mmfor
mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor" mmmeans mmthat mmif mmyou
mmhave mmno mmevidence mmof mmillness, mma mmdoctor mmwill mmgive mmyou mma mmclean
mmbill mmof mmhealth. mm- mmAnswerfalse
The mmstatement mmis mmfalse. mm"Absence mmof mmevidence mmof mmillness mmis
mmnecessary mmfor mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor" mmonly
mmmeans mmthat mmlack mmof mmevidence mmof mmillness mmis mma mmprecondition mmfor
mmreceiving mma mmclean mmbill mmof mmhealth mmfrom mma mmdoctor, mmbut mmmay mmnot
mmguarantee mmit.
Select mmwhich mmof mmthe mmfollowing mmexplains mmthe mmconditional mmcorrectly.
mmThere mmmay mmbe mmmore mmthan mmone mmcorrect mmchoice. mmFor mmfull mmcredit
mmon mmthis mmquestion, mmyou mmmust mmcheck mmall mmcorrect mmexplanations mmand
mmnone mmof mmthe mmincorrect mmones.
(Precisely mmwhen mmmeans mmif mmand mmonly mmif. mmDisjunctions mmare mminclusive.)
a. mmThe mmconditional mmis mmfalse mmprecisely mmwhen mmthe mmpremise mmis mmtrue
mmand mmthe mmconclusion mmis mmfalse.
b. mmThe mmconditional mmis mmtrue mmprecisely mmwhen mmthe mmpremise mmis mmfalse
mmor mmthe mmconclusion mmis mmtrue.
c. mmThe mmconditional mmis mmfalse mmprecisely mmwhen mmthe mmpremise mmis mmfalse
mmand mmthe mmconclusion mmis mmtrue.
d. mmThe mmconditional mmis mmtrue mmprecisely mmwhen mmthe mmpremise mmis mmtrue mmor
mmthe mmconclusion mmis mmfalse. mm- mmAnswera mmand mmb
The mmconditional mmis mmbest mmunderstood mmas mma mmpromise: mmif mmA mmhappens,
mmthen mmB mmwill mmhappen. mmThe mmonly mmway mma mmpromise mmcan mmbe
mmconsidered mmbroken mmis mmif mmA mmhappened mmbut mmB mmdid mmnot mmhappen. mmIn
mmall mmother mmcases, mmwe mmconsider mmthe mmpromise mmkept, mmincluding mmthe
mmcase mmwhere mmA mmdid mmnot mmhappen. mmThen, mmthe mmpromise mmwas mmnever
mmput mmto mmthe mmtest, mmand mmis mmconsidered mmkept mmby mmdefault.
