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)