Lecture No. 8
Boolean Algebra and Logic Simplification
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Boolean Algebra - Digital Logic Design - Lecture Slides, Slides of Digital Logic Design and Programming
Boolean Algebra, TTL Series, CMOS Series, Boolean Addition and Multiplication, Laws of Boolean Algebra, Rules of Boolean Algebra, Theorems of Boolean Algebra, Logic Circuits are main points of this lecture.
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2011/2012
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Lecture No. 8
Boolean Algebra and Logic Simplification
Recap
Operational Characteristics
DC Supply Voltage
Noise Margin
Power Dissipation
Frequency Response
Fan Out
CMOS Series
74HC 74AC 74AHC Propagation Delay (ns) 18 5 3. Power Dissipation (mW) Static 0.00275 0.0055 0. Power Dissipation (mW) at 100KHz 0.0625 0.08 0. Speed-Power product (pJ) at 100KHz 1.125 0.4 0. Max. Clock Rate (MHz) 50 160 170
74LV 74LVC 74ALVC Propagation Delay (ns) 9 4.3 3 Power Dissipation (mW) Static 0.0016 0.0008 0. Max. Clock Rate (MHz) 90 100 150
Boolean Algebra
Variable
Complement
Literal
Boolean Addition
Sum of literals
Sum term = 1 if any literal = 1
Sum term = 0 if all literals = 0
A + B A + B A +B + C
Boolean Multiplication
Product of literals
Product term = 1 if all literals = 1
Product term = 0 if any one literal = 0
A. B (^) A.B (^) A .B.C
Commutative Law
Commutative Law for Addition
A + B = B + A
Commutative Law for Multiplication
A.B = B.A
A B
A + B B A + B A
A B
A.B B A.B A
Associative Law
Associative Law for Addition
A + (B + C) = (A + B) + C
A
B
A+(B+C)
C
B+C
A B C
A+B (A+B)+C
Distributive Law
A.(B + C) = A.B + A.C
A B
A.(B+C)
C
B+C
A B
C
A.B A A.B+A.C A.C
Rules of Boolean Algebra
1. A + 0 = A
2. A + 1 = 1
3. A.0 = 0
4. A.1 = A
5. A + A = A
6. A + = 1
7. A.A = A
8. A. = 0
9. = A
10. A + A.B = A
11. A + = A + B
12. (A+B).(A+C)
= A+B.C
A
A
A
A. B
Demorgan’s Theorems
Any number of variables
Combination of variables
X. Y.Z = X+ Y + Z
X + Y+ Z = X.Y. Z
( A +B.C).(A.C +B) = (A +B.C) +(A.C +B )
= A .( B.C)+ (A.C). B = A^ .(B^ + C)+ (A +C).B = A .B + A.C + A.B+B. C = A .B + A.C+B. C
Boolean Analysis of Logic Circuits
Boolean Algebra provides concise way to
represent operation of a logic circuit
Complete function of a logic circuit can be
determined by evaluating the Boolean
expression using different input
combinations
Boolean Analysis of Logic Circuits
Inputs Output Inputs Output A B C D F A B C D F 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0
Simplification using Boolean Algebra