Higher Engineering Mathematics

BOOLEAN ALGEBRA AND LOGIC CIRCUITS 101

A

11.4 De Morgan’s laws

De Morgan’s laws may be used to simplifynot-
functions having two or more elements. The laws
state that:

A+B=A·B and A·B=A+B

and may be verified by using a truth table (see
Problem 11). The application of de Morgan’s laws
in simplifying Boolean expressions is shown in
Problems 12 and 13.

Problem 11. Verify thatA+B=A·B

A Boolean expression may be verified by using a
truth table. In Table 11.9, columns 1 and 2 give all
the possible arrangements of the inputsAandB. Col-
umn 3 is theor-function applied to columns 1 and 2
and column 4 is thenot-function applied to column

3. Columns 5 and 6 are thenot-function applied to
   columns 1 and 2 respectively and column 7 is the
   and-function applied to columns 5 and 6.

Table 11.9

1 2 3 4 5 6 7

A B A+B A+B A B A·B

0 0 0 1 1 1 1

0 1 1 0 1 0 0

1 0 1 0 0 1 0

1 1 1 0 0 0 0

Since columns 4 and 7 have the same pattern of 0’s

and 1’s this verifies thatA+B=A·B.

Problem 12. Simplify the Boolean expression

(A·B)+(A+B) by using de Morgan’s laws and

the rules of Boolean algebra.

Applying de Morgan’s law to the first term gives:

A·B=A+B=A+B sinceA=A

Applying de Morgan’s law to the second term gives:

A+B=A·B=A·B

Thus, (A·B)+(A+B)=(A+B)+A·B

Removing the bracket and reordering gives:
A+A·B+B

But, by rule 15, Table 11.8,A+A·B=A. It follows

that:A+A·B=A

Thus: (A·B)+(A+B)=A+B

Problem 13. Simplify the Boolean expression

(A·B+C)·(A+B·C) by using de Morgan’s

laws and the rules of Boolean algebra.

Applying de Morgan’s laws to the first term gives:

A·B+C=A·B·C=(A+B)·C

=(A+B)·C=A·C+B·C

Applying de Morgan’s law to the second term gives:

A+B·C=A+(B+C)=A+(B+C)

Thus (A·B+C)·(A+B·C)

=(A·C+B·C)·(A+B+C)

=A·A·C+A·B·C+A·C·C

+A·B·C+B·B·C+B·C·C

But from Table 11.8,A·A=AandC·C=B·B= 0

Hence the Boolean expression becomes:

A·C+A·B·C+A·B·C

=A·C(1+B+B)

=A·C(1+B)

=A·C

Thus: (A·B+C)·(A+B·C)=A·C

Now try the following exercise.

Exercise 48 Further problems on simpli-

fying Boolean expressions using de Morgan’s

laws

Use de Morgan’s laws and the rules of Boolean

algebra given in Table 11.8 to simplify the

following expressions.

1. (A·B)·(A·B)[A·B]


2. (A+B·C)+(A·B+C)[A+B+C]


3. (A·B+B·C)·A·B [A·B+A·B·C]


4. (A·B+B·C)+(A·B) [1]


5. (P·Q+P·R)·(P·Q·R)[P·(Q+R)]