082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

188 Chapter 5 Discrete mathematics
Solution
BylabellingintermediatepointsinthecircuitweseethatX =A·B +C.Inorderto
obtain the truth table we form all possible combinations ofA,BandC, followed by
A·B,CandfinallyX =A·B+C.ThecompletecalculationisshowninTable5.13.
Table5.13
The truthtableforFigure5.14.
A B C A·B C X=A·B+C
A
B
C
Figure5.14
Circuit forEngineeringapplication 5.1.
X
1 1 1 1 0 1
1 1 0 1 1 1
1 0 1 0 0 0
1 0 0 0 1 1
0 1 1 0 0 0
0 1 0 0 1 1
0 0 1 0 0 0
0 0 0 0 1 1
Whatwewouldnowliketobeabletodoiscarryoutthereverseprocess:thatis,startwith
a truth table and find an appropriate Boolean expression so that the required electronic
device can beconstructed.
Example5.10 Given inputsA,BandC, find a Boolean expression forX as given by the truth table in
Table 5.14.
Solution To find an equivalent Boolean expression the procedure is as follows. Look down the
rows of the truth table and select those with an r.h.s. equal to 1. In this example, there
arefivesuchrows:1,2,4,6and8.Eachoftheserowsgivesrisetoatermintherequired
Booleanexpression.Eachtermisconstructedsothatithasavalue1fortheinputvalues
of that row. For example, for the input values of row 1, that is 1, 1, 1, we findA·B·C
has the value 1, whereas for the input values of row 2, that is 1, 1, 0, we findA·B·C
has the value 1. Carrying out this process for the other rows we find that the required
expression is
Table5.14
Thetruthtable fora
systemwithinputsA,B
andC andan outputX.
A B C X
1 1 1 1
1 1 0 1
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 1
0 0 1 0
0 0 0 1
X =A·B·C+A·B·C+A·B ·C +A·B·C +A ·B·C (5.1)
thatis,adisjunctionofterms,eachtermcorrespondingtooneoftheselectedrows.This
important expression is known as a disjunctive normal form (d.n.f.). We note that the
truthtableisthesameasthatofEngineeringapplication5.1whichhadBooleanexpression
(A·B)+C. The d.n.f. wehave justcalculated, whilecorrect,isnot the simplest.
Moregenerally,tofindtherequiredd.n.f.fromatruthtablewewritedownanexpression
ofthe form
( )+( )+···+( )
whereeachtermhasthevalue1fortheinputvaluesofthatrow.Wecouldnowconstruct
an electronic circuit corresponding to Equation (5.1) using a number of AND and OR
gatestogetherwithinverters,anditwoulddotherequiredjobinthesensethatthedesired
truth table would beachieved.