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

5.4 Boolean algebra 185
A
B
A . B
Figure5.10
ANOT gate in serieswith an AND gate.
A
B
Figure5.11
Symbol foraNAND gate.
F = A . B
F = A . B
Table5.7
Thetruthtable foraNAND gate.
A B F =A ·B
1 1 0
1 0 1
0 1 1
0 0 1
5.3.5 TheNANDgate
This gate is logically equivalent to a NOT gate in series with an AND gate as shown in
Figure5.10.ItisrepresentedbythesymbolshowninFigure5.11andhasthetruthtable
definedby Table 5.7.
Althoughwehaveonlyexaminedgateswithtwoinputsitispossibleforagatetohave
morethantwo.Forexample,theBooleanexpressionforafour-inputNANDgatewould
beF=A·B·C ·D while that of a four-input OR gate would beF = A +B+C +D,
whereA,B,C andDare the inputs, andF is the output. Logic gates form the building
blocks formore complicated digital electronic circuits.
5.4 BOOLEANALGEBRA
SupposeAandBarebinarydigits,thatis1or0.These,togetherwiththelogicalconnectives
+and · andalsothecomplementNOT,formwhatisknownasaBooleanalgebra.
The quantities A and B are known as Boolean variables. Expressions such as A +B,
A·BandAareknownasBooleanexpressions.MorecomplexBooleanexpressionscan
bebuiltupusingmoreBooleanvariablestogetherwithcombinationsof +, · andNOT;
forexample, wecan draw up a truth table forexpressions such as (A·B) + (C ·D).
Weshallrestrictourattentiontothelogicgatesdescribedinthelastsectionalthough
the techniques of Boolean algebra are more widely applicable. A Boolean variable can
only take the values 0 or 1. For our purposes a Boolean algebra is a set of Boolean
variables with the two operations · and +, together with the operation of taking the
complement, for which certainlaws hold.
5.4.1 LawsofBooleanalgebra
ForanyBooleanvariablesA,B,C,wehavethelawsinTable5.8.Fromtheseitispossible
toprovethelawsgiveninTable5.9.Youwillnoticethattheselawsareanalogoustothose
ofsetalgebraifweinterpret +as ∪, · as ∩,1astheuniversalset E,and0astheempty
set ◦/. In ordinary algebra, multiplication takes precedence over addition. In Boolean
algebra · takes precedence over +. So, for example, we can write the first absorption
lawwithout brackets, thatis
A+A·B=A
Similarly, the first minimization law becomes
A·B+A·B =A
We shall follow thisruleof precedence fromnow on.