Lect 04Number System, Gates, Boolean
AlgebraAlgebra
CS221: Digital Design
Dr. A. SahuDept of Comp. Sc. & Engg.Dept of Comp. Sc. & Engg.
Indian Institute of Technology Guwahati
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Outline• Gates in Digital System
Basic Gates (AND OR & NOT)– Basic Gates (AND, OR & NOT)– Universal Gates (NAND & NOR)Others : XOR XNOR– Others : XOR, XNOR
• Boolean Algebra–Axioms
B l F ti• Boolean Functions
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Boolean AlgebraBoolean Algebra
How to prove 2+2=5?
29 9 9 92 2 4 ( 4 )2 2 2 2
+ = − + = − +
We know 2+2=4
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2 2 2 29 9 91 6 2 . 4 . ( )2 2 2
= − + +
2 2
2 2 29 9 9 92 0 ( ) 2 5 4 5 ( )2 2 2 2
= − + + = − + +
2 2 2
2 2 2 29 9 9 9 95 2 . 4 . ( ) ( 5 )2 2 2 2 2
= − + + = − +2 2 2 2 2
9 95 52 2
= − + = Where is the mistake?4√x2=x is true only when x≥0
Boolean Algebra• Computer hardware using binary circuit
greatly simply designg y p y g• George Boole (1813-1864): developed a
th ti l t t i 1847mathematical structure in 1847– To deal with binary operations with just two
values • Binary circuits: To have a conceptual y p
framework to manipulate the circuits algebraicallyalgebraically– Claude Shannon : 1937, Master Thesis
Basic Gates in Binary CircuitBasic Gates in Binary Circuit• Element 0 : “FALSE”. Element 1 : “TRUE”.• ‘+’ operation “OR”,‘*’ operation “AND” and ’
operation “NOT”.
OROR 00 11 ANDAND 00 11 NOTNOT
00 00 11 00 00 00 00 11
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11 11 11 11 00 11 11 00
OR GateOR Gate• ‘+’ operation “OR”
X Y R=X OR YR= X + YR X + Y
0 0 0X
Y
R=X+Y
OROR 00 11
0 1 1
1 0 1OROR 00 1100 00 11 1 1 1
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11 11 11 1 + Y = 1
AND GateAND Gate• ‘*’ operation “AND”
X Y R=X AND YR= X * Y
0 0 0
0 1 0
X
Y
R=X*Y
0 1 0
1 0 0ANDAND 00 111 1 100 00 00
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11 00 11 0 * Y = 0
NOT GateNOT Gate• ‘ operation “NOT” or use BAR• R= X R=X’
R= NOT XX R NOT X0 1X
R=X’
1 0
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Boolean Algebra Defined
• Boolean Algebra B : 5-tuple
{B + * ’ 0 1}{B, +, *, ’, 0, 1}• + and * are bi operators• + and * are binary operators,
• ‘ is a unary operator• is a unary operator.
Boolean Algebra Defined• Axiom #1: Closure
If a and b are BooleanIf a and b are Boolean
(a + b) and (a * b) are Boolean.( ) ( )• Axiom #2: Cardinality/Inverse
if a is Boolean then a’ is Boolean
• Axiom #3: CommutativeAxiom #3: Commutative(a + b) = (b + a)(a * b) = (b * a)
Boolean Algebra DefinedA i #4 A i i If d b B l•Axiom #4: Associative : If a and b are Boolean
(a + b) + c = a + (b + c)( ) ( )(a * b) * c = a * (b * c)
•Axiom #6: Distributive a * (b + c) = (a * b) + (a * c)a + (b * c) = (a + b) * (a + c)a + (b c) = (a + b) (a + c)2nd one is Not True for Decimal numbers System5+(2*3) ≠ (5+2)*(5+3)11 ≠ 56
Boolean Algebra DefinedBoolean Algebra Defined
•Axiom #5: Identity Element :•Axiom #5: Identity Element : • B has identity to + and *
00 is identity element for + : a + 0 = a 1 is identity element for * : a * 1 = a1 is identity element for : a 1 a
•Axiom #7: Complement ElementAxiom #7: Complement Elementa + a' = 1a * a' = 0
Terminology• Juxtaposition implies *
operation:operation:ab = a * b
• Operator order of precedence is: ( ) ’ *( ) > ’ > * > +
+b +(b* ) ≠ ( +b)*a+bc = a+(b*c) ≠ (a+b)*cab’ = a(b’) ≠ (a*b)’ab a(b ) ≠ (a b)
Named TheoremsIdempotentIdempotent a + a = aa + a = a a * a = aa * a = aB d dB d d + 1 1+ 1 1 * 0 0* 0 0BoundednessBoundedness a + 1 = 1a + 1 = 1 a * 0 = 0a * 0 = 0AbsorptionAbsorption a + (a*b) = a a*(a+b) = a( ) ( )AssociativeAssociative ((a+ba+b)+c=)+c=
((bb ))(a*b)*c=(a*b)*c=*(b* )*(b* )a+(a+(b+cb+c)) a*(b*c)a*(b*c)
InvolutionInvolution (a’)’ = a(a’)’ = aInvolutionInvolution (a ) = a(a ) = a
DeMorgan’sDeMorgan’s (a+b)’ = a’ * b’(a+b)’ = a’ * b’ (a*b)’=a’ + b’(a*b)’=a’ + b’DeMorgan sDeMorgan s (a+b) a b(a+b) a b (a b) a + b(a b) a + b
Simplification Theorem • Uniting : XY XY' XXY + XY' = X X(Y+Y’)=X.1=X
(X + Y)(X + Y') = X XX+XY’+YX+YY’=X+X(Y+Y’)+0=X( )( )
• Absorption: X + XY = X X(1+Y)=X.1=X
X(X + Y) = X XX+XY X+XY XX(X + Y) = X XX+XY=X+XY=X
• Adsorptionp(X + Y')Y = XY, XY' + Y = X + Y XY+YY’=XY+0=XY
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Principle of Duality• Dual of a statement S is obtained
B i h i * d
Principle of Duality
• By interchanging * and + • By interchanging 0 and 1
• Dual of (a*1)*(0+a’) = 0 is (a+0)+(1*a’) = 1
Duality examplesDuality examples• x + 0 = x x.1=x• z + x’=1 x.x’=0
’ ( ’ )• A+B’C A. (B’+C)• A’B’+AB (A’+B’) (A+B)A B +AB (A +B ).(A+B)
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Consensus TheoremConsensus Theorem• XY + X'Z + YZ = XY + X'ZXY + X Z + YZ XY + X Z
XY + X'Z + YZ = xy + x'z + (x + x')yz xy + x z + (x + x )yz= xy + x'z + xyz + x'yz= xy + xyz + x'z + x'yz xy xyz x z x yz= xy(1 + z) + x'z(1 + y)= xy + x'zy
Consensus (collective opinion) of X.Y and X'.Z is Y.Z • (X + Y)(X' + Z)(Y + Z) = (X + Y)(X' + Z)(X Y)(X Z)(Y Z) (X Y)(X Z)
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Shannon ExpansionShannon Expansion• F(A B)= A’ F(0 B) + A F(1 B)• F(A,B)= A . F(0,B) + A. F(1,B)Example: pF(A,B) = A’.B+A.B’
=A’.(1.B+0.B’)+A.(0.B+1B’)=A’B+AB’=A B+AB
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Shannon ExpansionShannon Expansion• F(X, Y, Z)= X . F(1,Y,Z) + X’. F(0, Y, Z)Example: XY+X’Z+YZ
=X. ( 1.Y+0.Z+YZ) + X’ ( 0.Y+1.Z+YZ)=X (Y+YZ)+X’(Z+YZ)X.(Y+YZ)+X (Z+YZ)=XY+XYZ+X’Z+X’YZXY+XYZ+X’Z+X’YZ+XYZ=XY+XYZ+X’Z+X’YZ+XYZ= XY(1+Z)+X’Z+YZ(X’+X)=XY+X’Z+YZ
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