LOGIC GATE

1. S.No. Contents
Page-
No
I
Introduction
II BOOLEAN
ALGEBRA
III Basic
Gates
IV OR
Gate
v AND
Gate

2. VI NOT
Gate
VII NOR
Gate
VII NAND
Gate
vIII EX-OR
Gate
IX EX-
NOR Gate
X experiment

3. In the modern world of electronics, the term Digital
is generally associated with a computer because the
term Digital is derived fromthe way computers
performoperation, by counting digits. For many years,
the application of digital electronics was only in the
computer system. Butnow-a-days, digitalelectronics is
used in many other applications
Signal
Signalcan be defined as a physical quantity, which
contains some information. It is a function of one or
more than one independent variables. Signals are of two
types.
 Analog Signal
 Digital Signal
Analog Signal
An analog signal is defined as the signal having
continuous values. Analog signal can have infinite
number of different values. In real world scenario, most
of the things observed in nature are analog. Examples
of the analog signals are following.
 Temperature
 Pressure
 Distance
KEY POINT
*Following are some of the
examples in which Digital
electronics is heavily
used.
 Industrial process
control
 Military system
 Television
 Communication
system
 Medical equipment
 Radar
 Navigation

4.  Sound
 Voltage
 Current & power
Graphical representation of Analog Signal
(Temperature)
The circuits that process the analog signals are called as analog circuits or system.
Examples of the analog system are following.
 Filter
 Amplifiers
 Television receiver
 Motor speed controller
Disadvantage of Analog Systems
 Less accuracy
 Less versatility
 More noise effect
 More distortion
 More effect of weather

5. Digital Signal
A digital signal is defined as the signal which has only a finite number of distinct
values. Digital signals are not continuous signals. In the digital electronic
calculator, the input is given with the help of switches. This input is converted into
electrical signal which have two discrete values or levels.
One of these may be called low level and another is called high level. The signal
will always be one of the two levels.
This type of signal is called digital signal. Examples of the digital signal are
following.
 Binary Signal
 Octal Signal
 Hexadecimal Signal
Graphical representation of the Digital Signal (Binary)

6. The circuits that process the digital
signals are called digital systems or
digital circuits. Examples of the digital
systems are following.
 Registers
 Flip-flop
 Counters
 Microprocessors
Number System &
Conversion
"A set of values used to represent
different quantitiesis known as Number
System". For example, a number system
can be used to represent the number of
students in a class or number of viewers
watching a certain TV program etc. The
digital computer represents all kinds of
data and information in binary numbers. It
includes audio, graphics, video, text and
numbers. The total number of digits used
in a number system is called its base or
radix. The base is written after the number
as subscript such as 51210.
Some important number systems are as
follows.
Advantage of Digital
Systems
 More accuracy
 More versatility
 Less distortion
 Easy communicate
 Possible storage of information
 Comparison of Analog
and Digital Signal
S.N. Analog Signal Digital Signal
1
Analog signal has
infinite values.
Digital signal
has a finite
number of
values.
2
Analog signal has a
continuous nature.
Digital signal
has a discrete
nature.
3
Analog signal is
generated by
transducers and
signal generators.
Digital signal is
generated by A
to D converter.
4
Example of analog
signal − sine wave,
triangular waves.
Example of
digital signal −
binary signal.
ADVANTAGE AND
COMPARISON

7.  Decimal number system
 Binary number system
 Octal number system
 Hexadecimal number system
The decimal number system is used in general. However, the computers use
binary number system. The octal and hexadecimal number systems are used in
the computer.
Decimal number System
The Decimal Number System consists of ten digits from 0 to 9. These digits can
be used to represent any numeric value. The base of decimal number system is
10. It is the most widely used number system. The value represented by
individual digit depends on weight and position of the digit.
Each number in this system consists of digits which are located at different
positions. The position of first digit towards left side of the decimal point is 0.
The position of second digit towards left side of the decimal point is 1. Similarly,
the position of first digit towards right side of decimal point is -1. The position
of second digit towards right side of decimal point is -2 and so on.
The value of the number is determined by multiplying the digits with the weight
of their position and adding the results. This method is known as expansion
method. The rightmost digit of number has the lowest weight. This digit is called
Least Significant Digit (LSD). The leftmost digit of a number has the highest

8. weight. This digit is called Most Significant Digit (MSD). The digit 7 in the number
724 is most significant digit and 4 is the least significant digit.
Example:
The weights and positions of each digit of the number 453 are as follows:
Position 2 1 0
Weights 102101 100
Face value4 5 3
The above table indicates that:
The value of digit 4 = 4x102 = 400
The value of digit 4 = 5x10 = 50
The value of digit 3 = 3x10 = 3
The actual number can be found by adding the values obtained by the digits as
follows:
400 + 50 + 3 =45310
Example:
The weights and positions of each digit of the number 139.78 are as follows.
Position 2 1 0 -1 -2
Weights 102101 100.10-1 10-2
Face Value1 3 9 7 8
The above table indicates that:
The value of digit 1 = 1x102 = 100
The value of digit 3 = 3x101 = 30

9. The value of digit 9 = 9x100 = 9
The value of digit 7 = 7x10-1 = 0.7
The value of digit 8 = 8x10-2 = 0.08
The actual number can be found by adding the values obtained by the digits as
follows:
100 + 30 + 9 + 0.7 + 0.8 = 139.78
Binary Number System
Digital computer represents all kinds of data and information in the binary
system. Binary Number System consists of two digits 0 and 1. Its base is 2. Each
digit or bit in binary number system can be 0 or 1. A combination of binary
numbers may be used to represent different quantities like 1001.
The positional value of each digit in binary number is twice the place value or
face value of the digit of its right side. The weight of each position is a power of
2.The place value of the digits according to position and weight is as follows:
Position3 2 1 0
Weights23 222120
Example: Convert 101112 decimal number
Position 2 1 0 -1 -2
Weights 102101 10010-1 10-2
Face Value1 3 9 7 8
101112 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20
= 1 x 16 + 0 + 1 x 4 + 1 x 2 + 1 x 1

10. = 16 + 0 + 4 2 + 1
= 2310
Example: Convert 101.1012
Position 2 1 0 -1 -2 -3
Face Value1 0 1 .1 0 1
Weight 242120 2-12-2 2-3
101.1012 = 1 x 22 + 0x21 + 1 x 20 + 1x 2-1 + 0 x 2-2 + 1
x 2-3
= 1 x 4 + 0 + 1 x 1 + ½ + 0 + 1/8
= 4 + 0 + 1 + 0.5 + 0.125
= 5.62510
Decimal to Binary conversion
The following technique, called the (Division – Remainder Technique) is easiest &
simple method used to convert decimal to binary numbers.
Step – 1
Divide the decimal number to be converted by the value of the new base. In this
case divide it by 2.
Step – 2
Record the remainder from Step – 1 as the rightmost digit.
Step – 3
Divide the quotient of the previous by the new base.
Step – 4

11. Record the remainder from Step – 3 as the next digit (to the left) of the new base
number.
Step – 5
Bottom to top sequence of remainder will be the required converted number.
Repeat Step – 3 & Step – 4, recording remainders from right to left, until the
quotient becomes less than the digit of new base so that it cannot be divided.
All these steps are performed in the following picture for converting decimal
numbers to binary. Isn't that simple & easy.

12. Binary system complements
As the binary system has base r = 2. So the two types of complements for the binary system are
2's complement and 1's complement.
1's complement
The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is
called as taking complement or 1's complement. Example of 1's Complement is as follows.

13. 2's complement
The 2's complement of binary number is obtained by adding 1 to
the Least Significant Bit (LSB) of 1's complement of the number.
2's complement = 1's complement + 1
Example of 2's Complement is as follows.
Boolean algebra
Boolean algebra is used to analyze and simplify the digital (logic)
circuits. It uses only the binary numbers i.e. 0 and 1. It is also
called as Binary Algebra or logical Algebra. Boolean algebra
was invented by George Boole in 1854.
Rule in Boolean Algebra
Following are the important rules used in Boolean algebra.
 Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
TOPICES-
-Rules in Boolean
algebra.
- Boolean Algebra
law.
1. Commutative
law
1.1- Associative
law
1.2- Distributive
law
1.3- AND law
1.4-OR law
2. INVERSION
law
3. Theorems &

14.  Complement of a variable is represented by an over bar (-). Thus, complement of variable
B is represented as . Thus if B = 0 then = 1 and B = 1 then = 0.
 O Ring of the variables is represented by a plus (+) sign between them. For example O
Ring of A, B, C is represented as A + B + C.
 Logical AND of the two or more variable is represented by writing a dot between them
such as A.B.C. Sometime the dot may be omitted like ABC.
Boolean Laws
There are six types of Boolean Laws.
Commutative law
Any binary operation which satisfies the following expression is referred to as commutative
operation.
Commutative law states that changing the sequence of the variables does not have any effect on
the output of a logic circuit.
Associative law
This law states that the order in which the logic operations are performed is irrelevant as their
effect is the same.
Distributive law
Distributive law states the following condition.
AND law

15. These laws use the AND operation. Therefore they are called as AND laws.
OR law
These laws use the OR operation. Therefore they are called as OR laws.
INVERSION law
This law uses the NOT operation. The inversion law states that double inversion of a variable
results in the original variable itself.
Important Boolean Theorems
Following are few important boolean Theorems.
Boolean function/theorems Description
Boolean Functions BooleanFunctionsandExpressions,K-MapandNANDGates
realization
De Morgan's Theorems De Morgan's Theorem1 and Theorem2
Boolean function:-

16. Boolean algebra deals with binary variables and logic operation. A Boolean
Function is described by an algebraic expression called Boolean expression
which consists of binary variables, the constants 0 and 1, and the logic operation
symbols. Consider the following example.
Here the left side of the equation represents the output Y. So we can state equation
no. 1
Truth Table Formation
A truth table represents a table having all combinations
of inputs and their corresponding result.
It is possible to convert the switching equation into a
truth table. For example, consider the following switching
equation.
The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth
table for this equation is shown in Table (a). The number of rows in the
truth table is 2n where n is the number of input variables (n=3 for the given
equation). Hence there are 23 = 8 possibleinput combination of inputs.
Truth Table Formation
A truth table represents a table having all combinations of inputs and their
corresponding result.
It is possible to convert the switching equation into a truth table. For
example, consider the following switching equation.

17. The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth table
for this equation is shown in Table (a). The number of rows in the truth table
is 2n where n is the number of input variables (n=3 for the given equation).
Hence there are 23 = 8 possible input combination of inputs.
Methods to Simplify a Boolean Function
The methods used for simplifying a Boolean function are as follows:
-map or K-map, and
Karnaugh-map or K-map

18. The Boolean theorems and the De-Morgan's theorems are useful in manipulating
the logic expression. We can realize the logical expression using gates. The number
of logic gates required for the realization of a logical expression should be reduced
to a minimum possible value by K-map method. This method can be done in two
different ways, as discussed below.
Sum of Products (SOP) Form
It is in the form of sum of three terms AB, AC, BC with each individual term is a
product of two variables. Say A.B or A.C etc. Therefore such expressions are known
as expression in SOP form. The sum and products in SOP form are not the actual
additions or multiplications. In fact they are the OR and AND functions. In SOP
form, 0 represents a bar and 1 represents an unbar. SOP form is represented by .
Given below is an example of SOP.

19. Product of Sums (POS) Form
It is in the form of product of three terms (A+B), (B+C), or (A+C) with each term is
in the form of a sum of two variables. Such expressions are said to be in the
product of sums (POS) form. In POS form, 0 represents an unbar and 1 represents
a bar. POS form is represented by .
]
Given below is an example of POS.

20. Logic gates are the basic building blocks of any digital system. It is an electronic
circuit having one or more than one input and only one output. The relationship
between the input and the output is based on a certainlogic. Based on this, logic
gates are named as AND gate, OR gate, NOT gate etc.
DE MORGAN'S
THEOREMS
De Morgan has suggestedtwotheoremswhichare extremelyusefulinBooleanAlgebra.The two
theoremsare discussedbelow.
Theorem 1

The left hand side (LHS) of this theorem represents a NAND gate with inputs A
and B, whereas the right hand side (RHS) of the theorem represents an OR gate
with inverted inputs.
Bubbled OR.

21. Table showingverificationof the De Morgan's firsttheorem:
Theorem 2

The LHS of this theorem represents a NOR gate with inputs A and B, whereas the
RHS represents an AND gate with inverted inputs.

22. Bubbled AND.
Table showingverificationof the De Morgan's secondtheorem:

24. Basic
Gates
AND Gate
A circuit which performs an AND operation is shown in figure. It has n input (n >= 2) and one
output.
Logic diagram

25. Truth Table
OR Gate
A circuit which performs an OR operation is shown in figure. It has n input (n >= 2) and one
output.
Logic diagram
Truth Table

26. NOT Gate
NOT gate is also known as Inverter. It has one input A and one output Y.
Logic diagram
Truth Table

27. NAND Gate: -
A NOT-AND operation is known as NAND operation. It has n input (n >= 2) and one output.
Logic diagram
Truth Table

28. NOR Gate
A NOT-ORoperation is known as NOR operation. It has n input (n >= 2) and one output.
Logic diagram
Truth Table

29. XOR Gate
XOR or Ex-OR gate is a special type of gate. It can be used in the half adder, full adder and
subtract or. The exclusive-OR gate is abbreviated as EX-OR gate or sometime as X-OR gate. It
has n input (n >= 2) and one output.
Logic diagram

30. Truth Table
XNOR Gate
XNOR gate is a special type of gate. It can be used in the half adder, full adder and subtract or.
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.
Logic diagram
Truth Table

31. AIM
To find truth table & Boolean
expression from given Diagram of
given Boolean expression.
GIVEN -

32. Boolean expression:
Y =
[(A.B)+B’].A
TRUTH TABLE:

33. OUTPUT:
A B B A.B (A.B)+B Y=
[(A.B+B)].A
0 0 1 0 1 0
0 1 0 0 0 0
1 0 1 0 1 1
1 1 0 1 1 1

34. Fig-1
I/P IP1 IP2 O/P
- L H L
I/P TP1 TP2 0/P
+ H H H
+ L H

35. - H L
I/P IP1 IP2 O/P
+ L H H
- H H

36. I/P IP1 IP2 O/P
- L L L
+ L L