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Table1040Solvingtheexample–step1
p
0
1
¬p
1
0
Inthesecondstep,wedeterminethevalueoftheexpression¬¬p.Alsointhis
caseweusetheinformationfromtheTable1.1,bearinginmind,however,that
thistimetheinputdataarethevaluesof¬p.Inthefirstrow,theexpression¬p
takesthevalue0,sothevalueof¬¬pis1.Inthesecondlinetheotherway
around.TheresultsarelistedintheTable1.5.
Table1050Solutionoftheexample
p
0
1
¬p
1
0
¬¬p
0
1
Itistimetodeterminethevaluesofthewholesentence.Thishastheformof
equivalence,theleftsideis¬¬p(thirdcolumnofTable1.5),therightside–p
(firstcolumnofTable1.5).Asitfollowsfromtheinformationprovided(for
example)inTable1.2,theequivalenceistrueifbothcomponentsentenceshave
thesametruthvalue.Inthefirstcaseboth¬¬pandparefalse,andtherefore
havethesametruthvalue.Thus,thelogicalvalueof¬¬p⇔pis1.Similarly,
inthesecondcase,thetwoformulae(¬¬pandp)aretrue.Thefinalresultsare
presentedinTable1.6.
Table1060Solutionoftheexample
p
0
1
¬p
1
0
¬¬p
0
1
¬¬p⇔p
1
1
Asyoucansee,inbothcases(i.e.inallthepossiblecases)thelogicalvalueof
theformulais1,soitisatautology.
Example2
Checkiftheformula(p∨q)∧(p⇔r)3(q∧r)isatautology.
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