17.5 Simplification of Boolean Functions – Pulse and Digital Circuits

17.5 SIMPLIFICATION OF BOOLEAN FUNCTIONS

Using the contents given in Table 17.21 we can now simplify Boolean functions, as illustrated below.

  1. To show that P + PQ = P

     

    P + PQ = P(1 + Q)

    We know that 1 + Q = 1. Therefore,

     

    P(1 + Q) = P(1)

    We also know that P.1 = P. Therefore,

     

    P + PQ = P

  2. To show that P + Q = P + Q
    We can write,

     

    P + Q = (P + )(P + Q) = 1(P + Q) = P + Q

    Therefore,

    P + Q = P + (P + )Q = P + (1)Q

    since P + P = 1

    = P + Q

    since 1.Q = Q

  3. To show that (P + Q)(P + R) = P + QR (P + Q)

     

    (P + R) = PP + PR + QP + QR

    Distributive law

    = P + PR + PQ + QR

    since P.P = P

    = P + PQ + QR

    since P + PR = P

    = P + QR

    since P + PQ = P

     

    Thus, it is possible to reduce the complexity of the logic circuits by simplifying the Boolean equations.

EXAMPLE

Example 17.1: Consider the logic circuit shown in Fig. 17.12(a) and write down the Boolean equation and present a simplified circuit.

FIGURE 17.12(a) The given logic circuit

Solution:

The Boolean function is:

 

f = PQ + QR(Q + R)

To simplify this, we apply the rules of Boolean Algebra.

PQ + QR(Q + R) = PQ + QQR + QRR

Distributive law

= PQ + QR + QR

since QQ = Q and RR = R

= PQ + QR

since QR + QR = QR

= Q(P + R)

 

The simplified circuit is as shown in Fig. 17.12(b).

FIGURE 17.12(b) The simplified circuit of Fig. 17.12(a)

EXAMPLE

Example 17.2: For the logic circuit shown in Fig. 17.13(a), write down the Boolean equation and draw a simplified circuit.

FIGURE 17.13(a) The given logic circuit

Solution:

The Boolean function for the Fig. 17.13(a) is:

f = P + PR + Q(P + R)

 

= P + PR + QP + QR

Distributive law

= P + PR + QR

since P + QP = P

= P + QR

since P + PR = P

The simplified circuit is as shown in Fig. 17.13(b).

FIGURE 17.13(b) The simplified circuit of Fig. 17.13(a)