Boolean Algebra &

Logic Gates

Common Postulates (Boolean Algebra)• Closure

N={1,2,3,4,5,…..}It is closed w.r.t +

i.e. a+b=cas a,b,cΣN

• Associative Law(x*y)*z = x*(y*z)for all x,y,z,ΣS

• Commutative Lawx*y = y*x for all x,yΣSx+y = y+x

x+y = y+xx.Y = y.x

Common Postulates (Boolean Algebra)• Identity Element

e*x = x*e = x x Σ Se+x = x+e = x0+x = x+0 = x1*x = x*1 = x

• Inversex*y = e a*1/a = 1x+y = ea+(-a) = 0

• Distributed Lawx*(y.z) = (x*y) . (x*z)x.(y+z) = (x.y) + (x.z)x+(y.z) = (x+y) . (x+z)

x+0 = 0+x = xx.1 = 1.x = x

x+x’ = 1x.x’ = 0

Boolean Algebra and Logic Gatesx y x.y x y x+y x x’

0 0 0 0 0 0 0 1

0 1 0 0 1 1 1 0

1 0 0 1 0 1

1 1 1 1 1 1

x y z Y+z x.(y+z) x.y x.z (x.y)+x.z

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

x.(y+z) = (x.y)+(x.z)

Postulates and Theorems of Boolean Algebra

Postulate 2 (a) x+0 = x (b) x.1 = x

Postulate 5 (a) x+x’ = 1 (b) x.x’ = 0

Theorem 1 (a) x+x = x (b) x.x = x

Theorem 2 (a) x+1 = 1 (b) x.0 = 0

Theorem3, involution (x’)’ = x

Postulate3, commutative (a) x+y = y+x (b) xy = yx

Theorem4, associative (a) x+(y+z)=(x+y)+z (b) x(yz) = (xy)z

Postulate4, distributive (a) x(y+z)=xy+xz (b) x+yz = (x+y)(x+z)

Theorem5, DeMorgan (a) (x+y)’ = x’y’ (b) (xy)’ = x’+y’

Theorem6, absorption (a) x+xy = x (b) x(x+y)=x

Theorems1a. x+x = x

x+x = (x+x).1 = (x+x)(x+x’) = x+xx’ =x+0 =x

1b. x.x = x (Remember Duality of 1a)x.x = xx+0 = xx+xx’ = x(x+x’) = x.1 =x

Theorems2a. x+1 = 1

x+1 =1.(x+1)

= (x+x’)(x+1)

= (x+x’)

= x+x’

= 1

2b. X.0 = 0 (Remember Duality of of 2a)

3. (x’)’ = xComplement of x = x’Complement of x’ = (x’)’ = x

6a x+xy = xx+xy = x.1+xy = x(1+y) = x.1 =x

6b. x(x+y) = x (Remember Duality of 6a)

Can also be proved using truth table method

x y xy x+xy

0 0 0 0

0 1 0 0

1 0 0 1

1 1 1 1

x=x+xy

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

(x+y)’ = x’y’ DeMorgan’s Theorem (xy)’ = x’ +y’ DeMorgan’s Theorem

Operator Precedence1.( )2.NOT3.AND4.OR

x y

xy’ xy x’y

x y

x y

z

x+(y+z)

x y

z

xy+xz

VENN DIAGRAM FOR TWO VARIABLES VENN DIAGRAM ILLUSTRATION X=XY+X

VENN DIAGRAM ILLUSTRATION OF THE DISTRIBUTIVE LAW

x’y’

x y

TRUTH TABLE FOR F1=xyz’, F2=x+y’z, F3=x’y’z+x’yz+xy’ and F4=xy’+x’z

x y z F1 F2 F3 F4

0 0 0 0 0 0 0

0 0 1 0 1 1 1

0 1 0 0 0 0 0

0 1 1 0 0 1 1

1 0 0 0 1 1 1

1 0 1 0 1 1 1

1 1 0 1 1 0 0

1 1 1 0 1 0 0

xy

z

F1

z

y

F2x

(a) F1 = xyz’(b) F2 = x+y’z

(c) F3 = x’y’z+x’yz+xy’

F3z

y

x

(c) F4 = xy’+x’z

F4

z

y

x

Implementation of Boolean Function with GATES

Algebraic Manipulations for Minimization of Boolean Functions(Literal minimization)

1. x+x’y = (x+x’)(x+y) = 1.(x+y)=x+y

2. x(x’+y) = xx’+xy = 0+xy=xy

3. x’y’z+x’yz+xy’= x’z(y’+y)+xy’= x’z+xy’

4. xy+x’z+yz (Consensus Theorem)=xy+x’z+yz(x+x’)=xy+x’z+xyz+x’yz=xy(1+z)+x’z(1+y)=xy+x’z

5. (x+y)(x’+z)(y+z)=(x+y)(x’+z)by duality from function 4

Complement of a Function

(A+B+C)’ = (A+X)’= A’X’= A’.(B+C)’= A’.(B’C’)= A’B’C’

(A+B+C+D+…..Z)’ = A’B’C’D’…..Z’ (ABCD….Z)’ = A’+B’+C’+D’+….+Z’Example using De Morgan’s Theorem (Method-1)F1 = x’yz’+x’y’zF1’ = (x’yz’+x’y’z)’ = (x+y’+z)(x+y+z’)F2 = x(y’z’+yz)F2’= [x(y’z’+yz)]’ = x’+(y+z)(y’+z’)

Example using dual and complement of each literal (Method-2)

F1 = x’yz’ + x’y’zDual of F1 = (x’+y+z’)(x’+y’+z)Complement F1’ = (x+y’+z)(x+y+z’)

F2 = x(y’z’+yz)Dual of F2=x+(y’+z’)(y+z]Complement =F2’= x’+ (y+z)(y’+z’)

Minterm or a Standard Productn variables forming an AND term provide 2n possible combinations, called minterms or standard products (denoted as m1, m2 etc.).Variable primed if a bit is oVariable unprimed if a bit is 1Maxterm or a Standard Sumn variables forming an OR term provide 2n possible combinations, called maxterms or standard sums (denoted as M1,M2 etc.).Variable primed if a bit is 1Variable unprimed if a bit is 0

MINTERMS AND MAXTERMS FOR THREE BINARY VARIABLES

MINTERMS MAXTERMS

x y z Term Designation Term Designation

0 0 0 x’y’z’ m0 x+y+z M0

0 0 1 x’y’z m1 x+y+z’ M1

0 1 0 x’yz’ m2 x+y’+z M2

0 1 1 x’yz m3 x+y’+z’ M3

1 0 0 xy’z’ m4 x’+y+z M4

1 0 1 xy’z m5 x’+y+z’ M5

1 1 0 xyz’ m6 x’+y’+z M6

1 1 1 xyz m7 x’+y’+z’ M7

FUNCTION OF THREE VARIABLES

x y z Function f1 Function f2

0 0 0 0 0

0 0 1 1 0

0 1 0 0 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

f1 = x’y’z+xy’z’+xyz =m1 + m4 + m7f2 = x’yz+xy’z+xyz’+xyz = m3 + m5 + m6 + m7

f1 = x’y’z+xy’z’+xyzf1’ = x’y’z’+x’yz’+x’yz+xy’z+xyz’f1 =(x+y+z)(x+y’+z)(x+y’+z’)(x’+y+z’) (x’+y’+z)

= M0.M2.M3.M5.M6= M0M2M3M5M6

f2 = x’yz+xy’z+xyz’+xyzf2’ = x’y’z’+x’y’z+x’yz’+xy’z’f2 = (x+y+z)(x+y+z’)(x+y’+z)(x’+y+z)

= M0 M1 M2 M4Canonical FormBoolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form.M3+m5+m6+m7 or M0 M1 M2 M4

MINTERMS AND MAXTERMS FOR THREE BINARY VARIABLES

Sum of Minterms (Sum of Products)Example: F = A+B’CF = A(B+B’)+B’C(A+A’)

= AB+AB’+AB’C+A’B’C= AB(C+C’)+AB’(C+C’)+AB’C+A’B’C= ABC+ABC’+AB’C+AB’C’+AB’C+A’B’C= A’B’C+AB’C’+AB’C+ABC’+ABC= m1+m4+m5+m6+m7

F(A,B,C)=(1,4,5,6,7)

ORing of term AND terms of variables A,B &CThey are minterms of the function

Product of Maxterms (Product of sums)Example: F = xy+x’zF = xy+x’zF = (xy+x’)(xy+z) distr.law (x+yz)=(x+y)(x+z)

= (x+x’)(y+x’)(x+z)(y+z)= (x’+y)(x+z)(y+z)= (x’+y+zz’)(x+z+yy’)(y+z+xx’)= (x’+y+z)(x’+y+z’)(x+z+y)(x+z+y’)(y+z+x)(y+z+x’)= (x+y+z)(x+y’+z)(x’+y+z)(x’+y+z’)= M0 M2 M4 M5F(x,y,z) = (0,2,4,5)

ANDing of terms Maxterms of the function (4 OR terms of variables x,y&z)

Conversion between Canonical FormsF(A,B,C) = (1,4,5,6,7)

sum of mintermsF’(A,B,C) = (0,2,3)

= m0+m2+m3F(A,B,C) = (m0+m2+m3)’

= m0’.m2’.m3’ = M0 M2 M3 = (0,2,3) Product of maxterms

SimilarlyF(x,y,z) = (0,2,4,5)F(x,y,z) = (1,3,6,7)

Standard FormsSum of Products (OR operations)F1 = y’+xy+x’yz’ (AND term/product term)

Product of Sums (AND operations)F2=x(y’+z)(x’+y+z’+w)

(OR term/sum term)Non-standard formF3=(AB+CD)(A’B’+C’D’)

Standard form of F3F3=ABC’D’ + A’B’CD

TRUTH TABLE FOR THE 16 FUNCTIONS OF TWO BINARY VARIABLES

x y F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Operator symbols

+ , ,

F0 = 0 F1 = xy F2 = xy’ F3 = x

F4 = x’y F5 = y F6 = xy’ +x’y F7= x +y

F8 = (x+y)’ F9 = xy +x’y’ F10 = y’ F11 = x +y’

F12 = x’ F13 = x’ + y F14 = (xy)’ F15 = 1

BOOLEAN EXPRESSIONS FOR THE 16 FUNCTIONS OF TWO VARIABLE

BOOLEAN OPERATOR NAME COMMENTS FUNCTIONS SYMBOL

F0 =0 NULL BINARY CONSTANT 0

F1=xy x.y AND x and y

F2=xy’ x/y inhibition x but not yF3=x transfer x F4=x’y y/x inhibition y but not xF5=y transfer yF6=xy’+x’y x y exclusive-OR x or y but not bothF7=x+y x+y OR x or yF8=(x+y)’ x y NOR not ORF9=xy+x’y’ x y *equivalence x equals y F10=y’ y’ complement not yF11=x+y’ x y implication if y then xF12=x’ x’ complement not xF13=x’+y x y implication if x then yF14=(xy)’ x y NAND not ANDF15=1 IDENTITY BINARY CONSTANT 1

• *Equivalence is also known as equality, coincidence, and exclusive NOR

• 16 logic operations are obtained from two variables x &y• Standard gates used in digital design are: complement,

transfer, AND, OR , NAND, NOR, XOR & XNOR (equivalence).

H and L LEVEL IN IC LOGIC FAMILIES

IC Family Voltage

Type Supply (V)

High-level voltage

(V)

Range Typical

Low-level voltage (V)

TTL Vcc=5

ECL VEE=-5.2

CMOS VDD=3--10

Positive Logic:

Negative Logic

2.4-5 3.5

-0.95- -0.7 -0.8

VDD VDD

Logic-1

Logic-0

0-0.4 0.2

-1.9-- -1.6 -1.8

0-0.5 0

Logic-0

Logic-1

Range Typical

TYPICAL CHARACTERISTICS OF IC LOGIC FAMILIES

IC Logic

Family

Fan out Power

Dissipation (mw)

Propagation delay (ns)

Noise Margin (v)

Standard TTL

Shottky TTL

Low power

Shottky TTL

ECL

CMOS

10

10

20

25

50

10

22

2

25

0.1

10

3

10

2

25

0.4

0.4

0.4

0.2

3

TTL basic circuit : NAND gateECL basic circuit: NOR gateCMOS basic circuit: Inverter to construct NAND/NOR

DIGITAL LOGIC GATESNAME GRAPHIC

SYMBOL

ALGEBRIC

FUNCTION

TRUTH

TABLE

AND F=XY X Y F

0 0 0

0 1 0

1 0 0

1 1 1

OR F=X+Y X Y F

0 0 0

0 1 1

1 0 1

1 1 1

XY

F

YX

F

NAME GRAPHIC

SYMBOL

ALGEBRIC

FUNCTION

TRUTH

TABLE

Inverter

F=X’

X F

0 1

1 0

Buffer

F=X

X F

0 0

1 1

X F

X F

NAND F=(XY)’

X Y F

0 0 1

0 1 1

1 0 1

1 1 0

X FY

NAME GRAPHIC

SYMBOL

ALGEBRIC

FUNCTION

TRUTH

TABLE

NOR F=(X+Y)’

X Y F

0 0 1

0 1 0

1 0 0

1 1 0

Exclusive-OR

(XOR)

F=XY’+X’Y

= X Y

X Y F

0 0 0

0 1 1

1 0 1

1 1 0

FYX

X FY

Exclusive-NOR

or

EquivalenceF=XY+X’Y’

=X Y

X Y F

0 0 1

0 1 0

1 0 0

1 1 1

FXY

Y (X Y) Z=(X+Y) Z’

Y

x(X+Y)’

=XZ’+YZ’

[Z+(X+Y)’]’

(Y+Z)’

(X ( Y Z)=X’(Y+ Z)

=X’Y+X’Z

[X+(Y+Z)’]’

Z

X

Z

Demonstrating the nonassociativity of the NOR operator

(X Y) Z X (Y Z)

XYZ

(X+Y+Z)’XYZ

(XYZ)’

(a) There input NOR gate (b) There input NAND gate

ABC

D

E

F=[(ABC)’. (DE)’]’=ABC+DE

(c) Cascaded NAND gates

Multiple-input AND cascaded NOR and NAND gates

XY

Z F=X Y Z

(a) Using two input gates

XYZ

(b) Three input gates

(b) Three input exclusive OR gates

TRUTH TABLE

X Y Z F 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0

XOR

XNOROdd function

Even function

F=X Y Z

IC DIGITAL LOGIC FAMILIESTTL Transistor- Transistor Logic

• Very popular logic family.• It has a extensive list of digital functions.• It has a large number of MSI and SSI devices, also has LSI devices.

ECL Emitter Coupled Logic• Used in systems requiring high speed operations.• It has a large number of MSI and SSI devices, also LSI devices.

MOS Metal-Oxide Semiconductor• Used in circuit requiring high component density• It has a large number of MSI and SSI devices, also LSI devices

(mostly)CMOS Complementary MOS

• Used in systems requiring low power consumption.• It has a large number of MSI and SSI devices, also has LSI devices.

I2L Integrated - Injection Logic• Used in circuit requiring high component density.• Mostly used for LSI functions

1 2 3 4 5 6 7

14 13 12 11 10 9 8

VCC

GND1 2 3 4 5 6 7

14 13 12 11 10 9 8

VCC

GND

Some Typical IC Gates

7400 Quadruple 2-input NAND gates

7404 Hex Inverters

TTL gates

16 15 14 13 12 11 10 9

1 2 3 4 5 6 7 8

VCC 2

VEE 2 (-5.2V)VCC 1

10107 Triple Exclusive – OR/ NOR gates

16 15 14 13 12 11 10 9

1 2 3 4 5 6 7 8

VCC 2

VCC 1VEE (-5.2V)

10102 Quadruple 2-Input NOR gate

Some Typical IC Gates

1 2 3 4 5 6

NC

7

Vss (GND)

NC

8910111213

VDD

14

(3-15 V)

C MOS

GATES

4002 dual 4 input NOR gates

NC

16

1

VDD

3 4 5 6 7 8 Vss

(GND)

91011121415

2

(3-15 V)

4050 Hex buffer

CMOS

GATES

NC

13

0

1 H

L

0

1

H

L

LOGIC

VALUE

SIGNAL

VALUE

LOGIC

VALUE

SIGNAL

VALUE

Negative LogicPositive Logic

Signal amplitude assignment and type of logic

X y z

L L H

L H H

H L H

H H L

TTL7400GATE

x

y

z

Gate block diagramGate block diagramTruth table in terms of

H and L

X y z

0 0 1

0 1 1

1 0 1

1 1 0

Truth table for positive logic

H=1, L=0

x

yz

Graphic symbol for positive logic NAND gate

X y z

1 1 0

1 0 1

0 1 1

0 0 1

Truth table for negative logic

L=1 H=0

xz

y

Graphic symbol for negative logic NOR gate

+ive logic NAND or -ive logic NOR

+ive logic NOR or -ive logic NAND

Same gate can function

DEMONSTRATION OF POSITIVE AND NEGATIVE LOGIC

Fan-out

Specifies the number of standard loads (the amount of current needed by

an input of another gate in the same IC family) that the output of a gate can

drive without impairing its normal operation. it is expressed by a number.

Power dissipation

It is the supplied power required to operate the gate. It is expressed in mw.

Propagation delay

It is the average transition delay time for a signal to propagate from input to

output when the binary signals change in value. It is expressed in ns.

Noise margin

It is the maximum noise voltage added to the input signal of a digital circuit

that does not cause an undesirable change in the circuit output. It is

expressed in volts (v).

Characteristics of IC logic families(parameters)