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Solution
Sofarwedeterminedthelogicalvaluesoftheformulae,knowingthevaluesof
thesimplestatementsbeingtheirelements.Nowwehavetodotheopposite.But
actuallyitallboilsdowntothesamescheme.Westartfromwritingallthe
possiblecombinationsofsimplestatements(inourexample:pandq),thenon
thatbasiswedeterminethevaluesoftheresultingformulae.Onlythe
conclusionswedrawinadifferentway.Solet'sdeterminethevaluesofthe
formulae.ThecalculationresultsarecontainedintheTable1.12(notethatdue
tothepresenceoftwosimplestatementswehavetoconsider2
2
=4cases).
Table10120Solutionoftheexample
p
0
0
1
1
q
0
1
0
1
¬q
1
0
1
0
p∨¬q
1
0
1
1
p3q
1
1
0
1
Timetoconclude.Sinceweknowthatp∨¬qandp3qaretrue,wehaveto
findthelinescorrespondingtotherespectivevalues(1inthelasttwocolumns)
inTable1.12.Thesearetherows1and4.Inthefirsttwocolumnswereadthat
correspondstothevaluesofstatementspandqequalto0and0(firstrow)or1
and1(fourthrow).
Sofinallygivenformulaearetruewhenbothstatementspandqarefalseorboth
aretrue.
EXERCISES
1.Checkifthefollowingformulaearetautologies:
a)p3p;
b)p⇔p;
c)p∧¬p⇔0;
d)p∨¬p⇔1;
e)p∧1⇔p;
f)p∨1⇔1;
g)p∧0⇔0;
h)p∨0⇔p;
i)
j)
k)¬(p∨q)⇔(¬q∧¬p);
l)
m)p∨(q∨r)⇔(p∨q)∨r;
n)p∧(q∧r)⇔(p∧q)∧r;
p∨q⇔q∨p;
p∧q⇔q∧p;
¬(p∧q)⇔(¬q∨¬p);
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