0195136047.pdf

278 DIGITAL BUILDING BLOCKS AND COMPUTER SYSTEMS

TABLE 6.1.3Basic Boolean Identities

Identity Comments

1.X+ 0 =X Identities 1–9 are basic to Boolean algebra

2.X+ 1 = 1

3.X+X=X

4.X+X ̄= 1

5.X· 0 = 0

6.X· 1 =X

7.X·X=X

8.X·X ̄= 0

9.X ̄ ̄=X

10.X+Y=Y+X Commutative

11.X·Y=Y·X Commutative

12.X+(Y+Z)=(X+Y)+Z Associative

13.X·(Y·Z)=(X·Y)·Z Associative

14.X·(Y+Z)=X·Y+X·Z Distributive

15.X+Y·Z=(X+Y)·(X+Z)

16.X+X·Y=X Absorption

17.X·(X+Y)=X

18.X·Y+X ̄·Z+Y·Z=X·Y+X ̄·Z Consensus

19.X+Y+Z=X ̄·Y ̄·Z ̄ DeMorgan

20.X·Y·Z=X ̄+Y ̄+Z ̄ DeMorgan

EXAMPLE 6.1.3

For the switching functionF=A

(

A ̄+B

)

, draw a corresponding set of logic blocks and write

the truth table.

Solution

A suitable connection of logic blocks is shown in Figure E6.1.3. Using the intermediate variables,

the truth table is as follows.

A

B

A

F = A (A + B)

A + B

Figure E6.1.3

ABA A+B F

001 1 0

011 1 0

100 0 0

110 1 1

EXAMPLE 6.1.4

Derive the Boolean function for the combinational network shown in Figure E6.1.4(a).