##### 17.5 SIMPLIFICATION OF BOOLEAN FUNCTIONS

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

- 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* - 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* - 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+QRRDistributive law

=

PQ+QR+QRsince

QandRR=R=

PQ+QRsince

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+QRDistributive law

=

P+PR+QRsince

P+QP=P=

P+QRsince

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)