This document discusses simplification of circuits through Demorgan's theorem and Karnaugh maps. It begins with an introduction and table of contents. It then covers Demorgan's theorem, proving it using truth tables. It also discusses simplification using Boolean laws and defines various Boolean logic terms. The document covers Karnaugh maps for 2, 3, and 4 variables and how to simplify using them. It also discusses logic gates like AND, OR, NOT, NAND, NOR, XOR and XNOR gates through their truth tables and diagrams. Finally, it covers DeMorgan's theorems, proving them using truth tables and logic gates.

1. Simplification of circuits
Ms.Mary Jacob, Asst.Professor
Kristu Jayanti College (Autonomous),Bangalore

2. CONTENTS Demorgan’s theorem-Statements, proof using truth tables. Simplification of
Boolean expressions using Boolean laws. Definition of product term, sum term,
minterm, maxterms, SOP, standard SOP, POS and Standard POS, Conversion of
Boolean expression to standard SOP and standard POS forms
Introduction
01
Karnaugh maps-Definitions, for 2, 3, and 4 variables, grouping of cells,
redundant groups and don’t care conditions; Simplification of 3 and 4 variable
Boolean expression using K-maps (SOP only)
03 Karnaugh maps
02 Demorgan’s theorem
Comparison with analog systems, Need and advantages of digital systems,
Noise and error corrections. Logic gates: AND, OR, NOT, NAND, XOR, NOR:
Assembly with discrete components, Definition, pin diagram, truth table and
timing diagram

3. Analog Electronics
It is a branch of electronics that deals with a continuously variable signal.
It’s widely used in radio and audio equipment along with other
applications where signals are derived from analog sensors before being
converted into digital signals for subsequent storage and processing.
Analog Circuits
It can be defined as a complex combination of op amps, resistors, caps,
and other basic electronic components. These circuits can be as simple as
a combination of two resistors to make a voltage divider or elegantly
built with many components.
There are two types of analog circuits namely passive and active,
where the former ones don’t consume any electrical power while
the latter ones do.

4. Digital Electronics
It is a field of electronics involving the study of digital signals and the
engineering of devices that use or produce them.
Digital Electronic Circuits
A digital circuit is a circuit where the signal must be one of two discrete
levels. Each level is interpreted as one of two different states (for
example, on/off, 0/1, true/false). Digital circuits use transistors to create
logic gates in order to perform Boolean logic.
Digital circuits are usually made from large assemblies of logic gates,
often packaged in integrated circuits. Complex devices may have simple
electronic representations of Boolean logic functions

5. Differences between Analog and Digital Electronics

6. Ø Data is represented in a digital system as a vector of binary variables
Ø Digital Systems can provide accuracy (dynamic range) limited only by
the number of bits used to represent a variable
Ø Digital systems are less prone to error than analog systems
Ø Data representation in a digital system is suitable for error detection and
correction
Ø Digital systems are designed in a hierarchical manner using re-useable
modules.
Characteristics of Digital System

7. Benefits of Digital over Analog System
Ø It is economical and easy to design.
Ø It is very well suited for both numerical and non-numerical information
processing.
Ø It has high noise immunity.
Ø It is easy to duplicate similar circuits and complex digital ICs are
manufactured with the advent of microelectronics Technology.
Ø Adjustable precision and easily controllable by Computer.
Disadvantages of Digital System
Ø It has low speed.
Ø There are need of converters, e.g., Analog to Digital (A/D) converter and
Digital to Analog (D/A) converter, because physical world is analog.

8. Logic Gates
Ø Logic gates are the basic building blocks of any digital system.
Ø Logic gate is an electronic circuit wich makes logic decisions.
Ø It is an electronic circuit having one or more than one input and only
one output.
Ø The relationship between the input and the output is based on a certain
logic.
Ø Logic gates are named as AND gate, OR gate, NOT gate
Truth Table
v The input-output relationship of the binary variables for each gate can
be represented in a tabular form in truth table

9. ü It has n input (n >= 2) and one output.
ü The output is "true" when both inputs are "true." Otherwise, the output is
"false." In other words, the output is 1 only when both inputs one AND
two are 1.
Logic Gate Truth Table
AND Gate - Logical Multiplication

10. AND Gate
I
C
7
4
0
8
IC 7408

11. AND Laws
The boolean algebraic laws can be written as:
Ø A.B=B.A
Ø A.B.C=(A.B).C = A.(B.C)
Ø A.1=A
Ø A.0=0
Ø A.A=A
Ø A(B+C)=A.B+ A.C

12. ü It has n input (n >= 2) and one output.
ü If both inputs are "false," then the output is "false." In other words, for
the output to be 1, at least input one OR two must be 1.
Logic Gate Truth Table
OR Gate - Logical Addition

13. OR Gate
IC 7432

14. OR Laws
The boolean algebraic laws can be written as:
Ø A+B=B+A
Ø A+B+C=(A+B)+C = A+(B+C)
Ø A+1=1
Ø A+0=A
Ø A+A=A

15. ü NOT gate is also known as Inverter. It has one input A and one output Y.
ü It reverses the logic state. If the input is 1, then the output is 0. If the
input is 0, then the output is 1.
Logic Gate Truth Table
NOT Gate

16. NOT Gate - Pin and Timing Diagram
IC 7404

17. NOT Laws
_ _
0=1 and 1=0
Using AND ,OR and NOT definitions the boolean identities can be verified as
=
A=A
_
A+A=1
_
A.A=0
_
A+AB=A+B

18. Basic Laws of Boolean Algebra
Laws of
Complementation
1 _
0=1
2 _
1=0
3 =
A=A
OR Laws
1 0+0=0
2 0+1=1
3 1+0=1
4 1+1=1
5 A+0=A
6 A+1=1
7 A+A=A
8 _
A+A=1
AND Laws
1 0.0=0
2 0.1=0
3 1.0=0
4 1.1=1
5 A.0=0
6 A.1=A
7 A.A=A
8 _
A.A=0

19. Basic Laws of Boolean Algebra
Associative Laws
1 A+(B+C)=(A+B)+C
2 A.(B.C)=(A.B).C
Commutative Laws
1 A+B=B+A
2 A.B=B.A
Distributive Laws
1 A.(B+C)=A.B+A.C
2 A+(B.C)=(A+B). (A+C)
3 _
A+ (A.B)=A+B

20. ü A NOT-AND operation is known as NAND operation. It has n input (n
>= 2) and one output.
ü The NAND gate operates as an AND gate followed by a NOT gate. It
acts in the manner of the logical operation "and" followed by negation.
The output is "false" if both inputs are "true." Otherwise, the output is
"true."
Logic Gate Truth Table
NAND Gate

21. NAND Gate - Pin and Timing Diagram

22. ü A NOT-OR operation is known as NOR operation. It has n input (n >= 2)
and one output.
ü The NOR gate is a combination OR gate followed by an inverter. Its
output is "true" if both inputs are "false." Otherwise, the output is "false."
Logic Gate Truth Table
NOR Gate

23. NOR Gate

24. ü The exclusive-OR gate is abbreviated as XOR or Ex-OR gate is a special
type of gate. It has n input (n >= 2) and one output.
ü The output is "true" if either, but not both, of the inputs are "true." The
output is "false" if both inputs are "false" or if both inputs are "true."
Logic Gate Truth Table
XOR Gate

25. XOR Gate - Pin and Timing Diagram

26. ü The exclusive-NOR gate is abbreviated as EX-NOR gate or sometime as
X-NOR gate. It has n input (n >= 2) and one output.It can be used in the
half adder, full adder and subtractor.
ü It is a combination XOR gate followed by an inverter. Its output is "true"
if the inputs are the same, and "false" if the inputs are different.
Logic Gate Truth Table
XNOR Gate

27. XNOR Gate - Pin and Timing Diagram

28. ü The logical symbol 0 and 1 are used for representing the digital input or
output.
ü The symbols "1" and "0" can also be used for a permanently open and
closed digital circuit.
ü The digital circuit can be made up of several logic gates. To perform the
logical operation with minimum logic gates, a set of rules were
invented, known as the Laws of Boolean Algebra.
Boolean Algebra

29. Ø Only two values(1 - high and 0 - low) are possible for the variable used
in Boolean algebra
Ø The overbar(-) is used for representing the complement variable
Ø The plus(+) operator is used to represent the ORing of the variables
Ø The dot(.) operator is used to represent the ANDing of the variables
Boolean Algebra - Rules

30. 1. Annulment Law
When the variable is AND with 0, it will give the result 0, and when the
variable is OR with 1, it will give the result 1
i.e.) B.0 = 0 B+1 = 1
Boolean Algebra - Properties
2. Identity Law
When the variable is AND with 1 and OR with 0, the variable remains the
same
i.e.) B.1 = B B+0 = B
3. Idempotent Law
When the variable is AND and OR with itself, the variable remains same
or unchanged
i.e.) B.B = B B+B = B

31. 4.Complement Law
When the variable is AND and OR with its complement, it will give the
result 0 and 1 respectively
i.e.) B.B' = 0 B+B' = 1
Boolean Algebra - Properties
5. Double Negation Law
This law states that, when the variable comes with two negations, the
symbol gets removed and the original variable is obtained
i.e.)((A)')' = A
6. Commutative Law
This law states that no matter in which order we use the variables. It means
that the order of variables doesn't matter in this law
i.e.) A.B = B.A A+B = B+A

32. 7.Associative Law
This law states that the operation can be performed in any order when the
variables priority is of same
i.e.)(A.B).C = A.(B.C) (A+B)+C = A+(B+C)
Boolean Algebra - Properties
8. Distributive Law
This law allows us to open up of brackets.
i.e.)A+(B.C) = (A+B).(A+C ) A.(B+C) = (A.B)+(A.C)
9. Absorption Law
This law allows us for absorbing the similar variables
i.e.)B+(B.A) = B B.(B+A) = B

33. De Morgan Law
Ø The operation of an OR and AND logic circuit will remain same if we
invert all the inputs, change operators from AND to OR and OR to
AND, and invert the output.
(A.B)' = A'+B' (A+B)' = A'.B'
Ø It states that the complement of logical OR of at least two Boolean
variables is equal to the logical AND of each complemented variable.
Boolean Algebra - Properties

34. Duality Theorem
This theorem states that the dual of the Boolean function is obtained by
interchanging the logical AND operator with logical OR operator and
zeros with ones. For every Boolean function, there will be a corresponding
Dual function.
Boolean Algebra - Properties
Group1 Group2
x + 0 = x x.1 = x
x + 1 = 1 x.0 = 0
x + x = x x.x = x
x + x’ = 1 x.x’ = 0
x + y = y + x x.y = y.x
x + y+z = x+y+ z x.y.z = x.y.z
x.y+z = x.y+ x.z x + y.z = x+y.x+z

35. The DeMorgan's theorems are used for mathematical verification of the
equivalency of the NOR and negative-AND gates and the negative-OR
and NAND gates.
DeMorgan’s First Theorem
DeMorgan’s First theorem proves that when two or more input
variables are AND’ed and negated, they are equivalent to the OR of the
complements of the individual variables. Thus the equivalent of the
NAND function will be a negative-OR function, proving that
A.B .C.......= A + B + C + ............
DeMorgan's theorems

36. Verifying DeMorgan’s First Theorem using Truth Table

37. DeMorgan’s First Law Implementation using Logic Gates

38. DeMorgan’s Second Theorem
The complement of an OR sum equals the AND product of the complements
DeMorgan’s Second theorem proves that when two or more input variables are
OR’ed and negated, they are equivalent to the AND of the complements of the
individual variables. Thus the equivalent of the NOR function is a negative-AND
function proving that
A+B+C+.............. = A . B . C ..........

39. Verifying DeMorgan’s Second Theorem using Truth Table

40. DeMorgan’s Second Law Implementation using Logic Gates

41. DeMorgan’s Law-3 & 4 variable input

42. NAND Gate- Why Universal gate??

43. NOR Gate- Why Universal gate??

44. THANK YOU