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

5.4 Boolean algebra 195
(d) A ·B·(A+B)
(e) A·(A +B)·B
(f) A·B·C+A·B·C
(g) A·B·C+A +B+C
8 Constructatruthtable showingA·BandA +B in
order to verifythe logicalequivalence expressed in
DeMorgan’s lawA·B=A +B.Carryout asimilar
exercise to verifyA+B=A·B.
9 LetB = 1 andthenB = 0 in the absorption laws,and
use the identitylaws to obtain (a)A +A =Aand
(b)A ·A =A.Verify your resultsusingtruthtables.
10 Derive Booleanexpressions andtruthtablesforthe
circuits shown in Figure5.22.
11 Simplify the following Booleanexpressions using
Booleanalgebra:
(a) A·B+A·B+B·C+A·B·C
(b) A·(C+A)+C·B+D+C+B·C+C·A
(c) A·B·C·D+A·B·C+A·B·C·D +
A ·B·C·D+A·B·C ·D
12 Thetruthvalues ofthe Booleanexpression,X,are
given in the followingtables. WriteX in disjunctive
A
B
C
D
E
A
B
(a)
X
normal form.Usethe laws ofBooleanalgebra to
simplifyyourexpressions.
(a)
(b)
A B X A B X
(c)
(d)
0 0 1
0 1 0
1 0 1
1 1 1
A B C X
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 1
1 1 1 1
A B C X
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
0 0 1
0 1 1
1 0 0
1 1 0
13 ExpressA·B +A·B usingonly conjunction (AND
gate) and negation (NOTgate).
C
D
X
E
F
Figure5.22
(b)
Solutions
1 (a) (A+B)·C (b) (A·B)·B
(c) (A·B)+B+C (d) A +B
2 (a) A ·B +A ·B which isthe same asA·B+A·B
(b) (A·B+A·B)·(B+C)+(A·B +A·B)·(B+C)
3 SeeFigureS.14.