Boolean Algebra & Logic Gates
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)