1

Boolean Algebra and Logic GatesBoolean Algebra and Logic Gates

EE 240 – Logic Design

Chapter 2

Sohaib Majzoub

2

Logic Functions: Logic Functions: Boolean AlgebraBoolean Algebra

INVERTER

X X’

X X’0 11 0

If X=0 then X’=1If X=1 then X’=0

OR

AB

C=A+B

A B C0 0 00 1 11 0 11 1 1

If A=1 OR B=1 then C=1 otherwise C=0

AB

C=A·B

A B C0 0 00 1 01 0 01 1 1

If A=1 AND B=1 then C=1 otherwise C=0

AND

3

Axioms of Boolean AlgebraAxioms of Boolean Algebra

If X ≠ 0 then X = 1 If X ≠ 1 then X = 0

If X = 0 then X’ = 1 If X = 1 then X’ = 0

0 . 0 = 0 1 + 1 = 1

1 . 1 = 1 0 + 0 = 0

0 . 1 = 1 . 0 = 0 0 + 1 = 1 + 0 = 1

4

Boolean expressions and logic circuitsBoolean expressions and logic circuits

Any Boolean expression can be implemented as a combination of AND, OR, and NOT

X = [A’.(C+D)]’+B.E

CD

C+D[A’.(C+D)]’ [A’.(C+D)]’+B.E

BE

B.E

A’.(C+D)

A’A

5

Basic TheoremsBasic Theorems

X+0 = X

X0

C=XX 0 C0 0 01 0 1

X+1 = 1

X1

C=1X 1 C0 1 11 1 1

X0

C=0

X·0 = 0

X 0 C0 0 01 0 0

X1

C=X

X·1 = X

X 1 C0 1 01 1 1

Identity Law

Null Element

6

Basic Theorems:Basic Theorems:Idempotent LawsIdempotent Laws

X+X = X

XX

C=XX X C0 0 01 1 1

XX

C=X

X·X = X

X X C0 0 01 1 1

7

Basic Theorems:Basic Theorems: Involution Law Involution Law

X

(X’)’=X

BC=X

X B C0 1 01 0 1

8

Basic Theorems:Basic Theorems:Laws of ComplementarityLaws of Complementarity

X+X’ = 1

XX’

C=1X X’ C 0 1 1 1 0 1

XX’

C=0

X·X’ = 0

X X’ C0 1 01 0 0

9

Expression Simplification using the Expression Simplification using the Basic TheoremsBasic Theorems

X can be an arbitrarily complex expression.

Simplify the following boolean expressions as much as you can using the basic theorems.

(A.B’ + D).E + 1 =(A.B’ + D).(A.B’ + D)’ =(A.B + C.D) + (C.D + A) + (A.B + C.D)’ =

10

Associative LawAssociative Law

(X+Y)+Z = X+(Y+Z)

X Y Z X+Y (X+Y)+Z Y+Z X+(Y+Z)0 0 0 0 0 0 00 0 1 0 1 1 10 1 0 1 1 1 10 1 1 1 1 1 11 0 0 1 1 0 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 1 1 1

XY

ZC

YZ

XC

11

Associative LawAssociative Law

(XY)Z = X(YZ)

X Y Z XY (XY)Z YZ X(YZ) 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1

XY

ZC

YZ

XC

12

First Distributive LawFirst Distributive Law

X(Y+Z) = XY+XZ

X Y Z Y+Z X(Y+Z) XY XZ XY+XZ 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1

13

Second Distributive LawSecond Distributive Law

X+YZ = (X+Y)(X+Z)

X Y Z YZ X+YZ X+Y X+Z (X+Y)(X+Z) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1

14

Second Distributive LawSecond Distributive Law(A different proof)(A different proof)

(X + Y)(X + Z) = X(X + Z) + Y(X + Z) (using the first distributive law)

= XX + XZ + YX + YZ (using the first distributive law)

= X + XZ + YX + YZ (using the idempotent law)

= X·1 + XZ + YX + YZ (using the operation with 1 law)

= X(1 + Z + Y) + YZ (using the first distributive law)

= X·1 + YZ (using the operation with 1 law)

= X + YZ (using the operation with 1 law)

15

Distributive LawDistributive Law(Another formula)(Another formula)

X.Y + Z.W = (X+Z).(X+W).(Y+Z).(Y+W)

X.Y + Z.W.V =(X+Z).(X+W).(X+V).(Y+Z).(Y+W).(Y+V)

Proof?

16

More TheoremsMore Theorems

XY + XY’ = XXY + XY’ = X(Y + Y’) = X·1 = X

X + XY = XX(1 + Y) = X·1 = X

(X + Y)(X + Y’) = X(X + Y)(X + Y’) = XX + XY’ + YX + YY’ = X + X(Y’ + Y) + 0 = X + X·1 = X

X(X + Y) = XX(X + Y) = XX + XY = X·1 + XY = X(1 + Y) = X·1 = X

Combining Theorem

Covering or Absorption Theorem

17

The Consensus TheoremThe Consensus Theorem

XY + X’Z + YZ = XY + X’Z

= XY + X’Z + (X + X’)YZ

= XY + X’Z + XYZ + X’YZ

= XY + XYZ + X’Z + X’YZ

= XY(1 + Z) + X’Z(1 + Y)

= XY·1 + X’Z·1

XY + X’Z + YZ = XY + X’Z + 1·YZ

= XY + X’Z

(X+Y)(X’+Z)(Y+Z) = (X+Y)(X’+Z)

18

First DeMorganFirst DeMorgan’’s s LawLaw

The complement of the sum is equal the product of the complements.

(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’ 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

Z

Y

X

XY Z X

YZ

Four Different Equivalents of NOR Gate

19

Second DeMorganSecond DeMorgan’’s s LawLaw

The complement of the product is equal the sum of the complements.

(XY)’ = X’ + Y’

ZXY

X Y XY (XY)’ X’ Y’ X’ + Y’0 0 0 1 1 1 10 1 0 1 1 0 11 0 0 1 0 1 11 1 1 0 0 0 0

Z

Y

X

Four Different Equivalents of NAND Gate X

Y Z XY

Z

20

Generalized DeMorganGeneralized DeMorgan’’s Lawss Laws

De Morgan’s laws generalize to n variables:

(X1 + X2 + X3 + ··· + Xn)’ = X1’X2’X3’ ··· Xn’

(X1X2X3 ··· Xn)’ = X1’ + X2’ + X3’ + ··· + Xn’

21

DeMorganDeMorgan’’s Law (example)s Law (example)

Express the complement f’(w,x,y,z) of the following expression in a simplified form.

f(w,x,y,z) = wx(y’z + yz’)

f’(w,x,y,z) = w’ + x’ + (y’z +yz’)’

= w’ + x’ + (y’z)’(yz’)’

= w’ + x’ + (y + z’)(y’ + z)

= w’ + x’ + yy’ + yz + z’y’ + z’z

= w’ + x’ + 0 + yz + z’y’ + 0

= w’ + x’ + yz + y’z’

22

Principle of DualityPrinciple of Duality

• Any theorem or identity in switching algebra remains true if and + are swapped.

• The dual of a boolean function F is called FD

• It apply to all theorems that we worked with so far

• FD(X1,X2,…,Xn,+,.,’) = F(X1,X2,…,Xn,.,+,’)

• Parentheses has to be added to clarify precedence rules before the conversion

23Principle of DualityPrinciple of Duality

• Example: DeMorgan’s Laws – the second law is the dual of the first law

(X+Y)’ = X’Y’ (X Y)’ = X’ + Y’

• Another Example: Covering Theorem

X + X Y = X

The dual is: X X + Y = X

X + Y = X WRONG!!! Why?

Use parentheses to avoid problems X+(X.Y) then X.(X+Y)

24

Positive and Negative LogicPositive and Negative Logic

Positive Logic: the higher voltage (+V) represents 1 and the lower voltage (0V) represents 0

Negative Logic: the higher voltage (+V) represents 0 and the lower voltage (0V) represents 1

25

Positive and Negative Logic (example)Positive and Negative Logic (example)

LogicGate

e2

e3

e1

eo

The same physical circuit implements different logicfunctions. The function implemented depends on the logic used to interpret the inputs and outputs.

e1 e2 e3 eo

0V 0V 0V 0V0V 0V +V 0V0V +V 0V 0V0V +V +V 0V+V 0V 0V 0V+V 0V +V 0V+V +V 0V 0V+V +V +V +V

+

Electric Voltages

e1 e2 e3 eo

0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1

+

Positive Logic

e1 e2 e3 eo

1 1 1 11 1 0 11 0 1 11 0 0 10 1 1 10 1 0 10 0 1 10 0 0 0

+

Negative Logic

26

NAND: A Functionally Complete Logic GateNAND: A Functionally Complete Logic Gate

A C=(A·A)’ = A’

A B C 0 0 1 0 1 1 1 0 1 1 1 0

NAND

We can create all three gates (AND, OR, NOT) from NAND

AB

C=(A·B)’

NAND as an Inverter

NAND as an ANDAB

(A·B)’

C= (A’.B’)’ = A+BNAND as an ORB

A’

B’

A

C= ((A.B)’)‘= A.B

27

Is NOR Functionally Complete?Is NOR Functionally Complete?

• Prove it yourself!

AB C = (A+B)’

A B C 0 0 1 0 1 0 1 0 0 1 1 0

28

Exclusive OR, X-OR, or XORExclusive OR, X-OR, or XOR

X Y = XY’ + X’YX Y C0 0 00 1 11 0 11 1 0

If X=1 OR Y=1, butnot both of them, then C=1

XY

C

X 0 =

X 1 =

X X =

X X’ =

Commutative Law:X Y = Y X

Associative Law:(X Y) Z= X ( Y Z) = X Y Z

Distributive Law:X(Y Z) = XY XZ

X

X’

0

1

29

Exclusive-OR (cont.)Exclusive-OR (cont.)

Complement Law:(X Y)’ = X Y’ = X’ Y

X Y X’ Y’ XY (XY)’ XY’ X’Y0 0 1 1 0 1 1 10 1 1 0 1 0 0 01 0 0 1 1 0 0 01 1 0 0 0 1 1 1

Algebraic Proof:(X Y)’ = (XY’ + X’Y)’

= (XY’)’(X’Y)’= (X’ + Y)(X + Y’)= X’X + X’Y’ + XY + YY’= 0 + X’Y’ + XY + 0

= X’ Y= X’Y’ + XY= XY + X’Y’ = X Y’

30

Equivalence GateEquivalence Gate

(X Y) = XY + X’Y’X Y C0 0 10 1 01 0 01 1 1

If X=Y then C=1,otherwise C=0

(X Y) = (X Y)’

XY

C

31

Switch Networks and Switch DesignSwitch Networks and Switch Design

• Basic Ideal Switch: simplest structure in a computing system is a switch

• Path exists between INPUT and OUTPUT if Switch is CLOSED or ON

• Path does not exist between INPUT and OUTPUT if SWITCH is OPEN or OFF

32

Switches in SeriesSwitches in Series

Truth Table

S1 S2 Path?

AND Configuration

33

Switches in ParallelSwitches in Parallel

Truth Table

S1 S2 Path?

OR Configuration

34

CMOS SwitchesCMOS Switches

• The idea is to use the series and parallel switch configurations to route signals in a desired fashion.

• Unfortunately, it is difficult to implement an ideal switch as given.

• Complementary Metal Oxide Semiconductor (CMOS) devices give us some interesting components.

(INPUT)

(OUTPUT) (INPUT)

(OUTPUT)

35

CMOS Transfer CharacteristicsCMOS Transfer Characteristics

nMOS when CLOSED• Transmits logic level 0 well• Transmits logic level 1 poorly

pMOS when CLOSED• Transmits logic level 1 well• Transmits logic level 0 poorly

36

CMOS Transmission GateCMOS Transmission Gate

37

CMOS Transmission GateCMOS Transmission Gate

38High Impedance High Impedance ZZ

• With switches, we can consider three states for an output:– Logic-0– Logic-1– High Impedance Z

• Path exists for Logic-0 and Logic-1 when the switch is CLOSED.

• High impedance is a state where the switch is OPEN.

39High Impedance High Impedance ZZ

• Another way of thinking of switches is as follows:– Path exists for Logic-0 and Logic-1 when the switch is

CLOSED, meaning that the impedance/resistance is small enough to allow ample flow of current.

– High impedance is a state where the switch is OPEN, meaning that the impedance/resistance is very large allowing nearly no current flow.

40

Inverter (NOT) NetworkInverter (NOT) Network

Pull-Up

Pull-Down

Pull-Up Pull-Down

A B

01

A B

01

1Z 0

Z

A B

01 0

1

Truth Table

Inverter (NOT) Gate

B = A’ = A A B

41

NAND NetworkNAND Network

Pull-Up Pull-Down

A B C

0 0

Truth Table

0 1

1 0

1 1

A B C

0 0

0 1

1 0

1 1

A B C

0 0

0 1

1 0

1 1

NAND (NOT AND) Gate

C = (A.B)’ = (AB)’ = ABAB

C

42

NOR NetworkNOR Network

Pull-Up Pull-Down

A B C

0 0 1

Truth Table

0 1 Z

1 0 Z

1 1 Z

A B C

0 0 Z

0 1 0

1 0 0

1 1 0

A B C

0 0 1

0 1 0

1 0 0

1 1 0

NOR (NOT OR) Gate

C = (A+B)’ = (A+B)’ = A+BA

BC

43

AND NetworkAND Network

C

NAND Inverter

Truth Table

A B C

0 0 0

0 1 0

1 0 0

1 1 1

AND Gate

C = A.B = ABAB

C

44

OR NetworkOR NetworkNOR Inverter

C

Truth Table

A B C

0 0 0

0 1 1

1 0 1

1 1 1

OR Gate

C = A+BA

BC

45XOR NetworkXOR Network

Truth Table

A B C

0 0 0

0 1 1

1 0 1

1 1 0

XOR (Exclusive OR) Gate

C = A B = AB’ + A’B

A

BC

46XNOR (Equivalence) NetworkXNOR (Equivalence) Network

C

Truth Table

A B C

0 0 1

0 1 0

1 0 0

1 1 1

XNOR or Equivalence Gate

C = A B = AB + A’B’

A

BC

A

BC