BOOLEAN ALGEBRA AND LOGIC CIRCUITS 99A
Figure 11.17- Table 11.7, column 6
[
A·B·C+A·B·C+A·B·C
+A·B·C; see Fig. 11.18]Figure 11.1811.2 Simplifying Boolean expressions
A Boolean expression may be used to describe a
complex switching circuit or logic system. If the
Boolean expression can be simplified, then the num-
ber of switches or logic elements can be reduced
resulting in a saving in cost. Three principal ways of
simplifying Boolean expressions are:(a) by using the laws and rules of Boolean algebra
(see Section 11.3),
(b) by applying de Morgan’s laws (see Section 11.4),
and
(c) by using Karnaugh maps (see Section 11.5).
11.3 Laws and rules of Boolean
algebraA summary of the principal laws and rules of
Boolean algebra are given in Table 11.8. The way in
which these laws and rules may be used to simplify
Boolean expressions is shown in Problems 5 to 10.Table 11.8
Ref. Name Rule or law
1 Commutative laws A+B=B+A
2 A·B=B·A
3 Associative laws (A+B)+C=A+(B+C)
4 (A·B)·C=A·(B·C)
5 Distributive laws A·(B+C)=A·B+A·C
6 A+(B·C)
=(A+B)·(A+C)
7 Sum rules A+ 0 =A
8 A+ 1 = 1
9 A+A=A
10 A+A= 1
11 Product A· 0 = 0
12 rules A· 1 =A
13 A·A=A
14 A·A= 0
15 Absorption A+A·B=A
16 rules A·(A+B)=A
17 A+A·B=A+BProblem 5. Simplify the Boolean expression:
P·Q+P·Q+P·QWith reference to Table 11.8: ReferenceP·Q+P·Q+P·Q
=P·(Q+Q)+P·Q 5
=P· 1 +P·Q 10
=P+P·Q 12Problem 6. Simplify
(P+P·Q)·(Q+Q·P)With reference to Table 11.8: Reference(P+P·Q)·(Q+Q·P)
=P·(Q+Q·P)
+P·Q·(Q+Q·P)5
=P·Q+P·Q·P+P·Q·Q
+P·Q·Q·P 5
=P·Q+P·Q+P·Q
+P·Q·Q·P 13
=P·Q+P·Q+P·Q+ 014
=P·Q+P·Q+P·Q 7
=P·(Q+Q)+P·Q 5
=P· 1 +P·Q 10
=P+P·Q 12