boolean algrebra and logic gates in short

◦ Logic gates
◦ Boolean algebra

INTRODUCTION:
 A logic gate is an electronic circuit/device which
makes logic decisions.
 Most logic gates are two inputs and one outputs.
 At any given moment, every terminal is in one of
the two binary conditions low (0) or high(1),
represented by different voltage levels.

 The logic state of a terminal can, and generally
does, change often as the circuit processes data.
 In most logic gates, the low state is approximately
0v, while the high state is approximately 5v.
 Logic gates are also called as switches. with the
advent of integrate circuits, switches have been
replaced by TTL circuit and CMOS circuits.
 symbolic logic uses values, variables and
operations.

TYPES OF LOGIC GATES:
The most common logic gates used are,
Basic gates
1.OR
2.AND
3.NOT
Universal gates
1.NAND
2.NOR
 X-OR or Exclusive-OR

 Logic gates have special symbols:
OR gate
X
Y
Z = X + Y
 And waveform behavior in time as follows:
X 0 0 1 1
Y 0 1 0 1
(AND) X · Y 0 0 0 1
(OR) X + Y 0 1 1 1
(NOT) X 1 1 0 0
X
Y
Z = X · Y
AND gate
X Z = X
NOT gate or
inverter

OR GATE:
 The OR gate has two or more inputs and one
output.
 Its output is true if at least one input is true.
SYMBOL:

 The OR operation may be defined as “Y equals A
OR B”.
Y=A+B
Where, the symbol ‘+’ indicates the OR concept.
 Each terminal may assume two possible values
either zero or one.

TRUTH TABLE:
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1

AND GATE:
 The AND gate is also a basic kind of digital circuit.
 It has also two or more inputs and one output.
SYMBOL:

 The AND operation for the output is defined as, “y
equals A AND B”.
Y=A.B
 Where ‘.’ symbol indicates AND operation.
 The output of the AND gate is one only when both
inputs are one.

TRUTH TABLE:
A B A+B
0 0 0
0 1 0
1 0 0
1 1 1

NOT GATE or Inverter Gate:
 A NOT gate is a basic gate that has one input and
one output.
SYMBOL:

 The NOT circuit serves to invert the polarity of any
input pulse apply to it.
 If A is the input then output “Y equals to NOT A or
Ā.
Y= Ā
Where, the bar symbol over A represents NOT or
compliment operation

NAND GATE:
 The NAND gate is known as an universal gate
because it can be used to realize all the three
basic functions of OR, AND & NOT gates.
 It is also called as NOT-AND gate.
SYMBOL:

 The Boolean expression for the NAND operation
is given by,
Y=A.B

TRUTH TABLE:
A B AB
0 0 1
0 1 1
1 0 1
1 1 0

NOR GATE:
 The NOR gate is also a universal gate and it is a
combination of a NOT and OR gates.
SYMBOL:

 The Boolean expression for NOR gate is given
by,
Y=A+B

TRUTH TABLE:
A B A+B
0 0 1
0 1 0
1 0 0
1 1 0

Exclusive OR or X-OR GATE:
 The X-OR gate is a logic gate having two inputs
with and single output.
SYMBOL:

 The Boolean expression for the X-OR gate is
given by,
Y=A+B
Where + indicates the exclusive OR operation and
in terms of expression it can be expanded as
Y=AB+AB
+
+

TRUTH TABLE:
A B AB+AB
0 0 0
0 1 1
1 0 1
1 1 0

ADVANTAGES OF LOGIC GATES:
 It is generally very easy to reliably distinguish
between logic 1 or logic 0.
 The simplest flip-flop is the RS which is made up
of two gates.
 K-map is also designed by using logic gates. That
simplification helps when you start to connect
gates to implement the functions.
 These gates are also used in TTL and CMOS
circuitary.

 Boolean Algebra derives its name from the
mathematician George Boole in 1854 in his book
“An investigation of the laws of taught”.
 Instead of usual algebra of numbers Boolean
algebra is the algebra of truth values 0 or 1.
 In order to fully understand this the relation
between the AND gate, OR gate & NOT gate
operations should be appreciated.

POSTULATES OF BOOLEAN ALGEBRA:
 The Boolean algebra has its own set of
fundamental laws which differ from the ordinary
algebra. They are,
OR laws:
 A+0=A
 A+1=1
 A+A=A
 A+Ā=1

AND laws:
 A.0=0
 A.A=A
 A.1=A
 A.Ā=0
NOT laws:
 0=1
 1=0
 If A=0 then Ā=1
 If A=1 then Ā=0
 Ā=A

Commutative law:
 A+B=B+A
 A.B=B.A
Associative laws:
 A+(B+C)=(A+B)+C
 A.(B.C)=(A.B).C
 (A+B)+(C+D)=A+B+C+D

Distributive laws:
 A.(B+C)=(A.B)+(A.C)
 (A+B).C=A.C+B.C
 A+ĀB=A+B
 A+B.C=(A+B).(A+C)
Absorptive laws:
 A+A.B=A
 A.(A+B)=A
 A.(Ā+B)=AB
Demorgan’s laws:
 A+B=A.B
 A.B=A+B

EXAMPLE:
(AB+C)(AB+D)=AB+CD
 AB.AB+AB.D+C.AB+C.D
 AB+ABD+ABC+CD {A.A=A}
 AB(1+D)+ABC+CD {1+A=1}
 AB+ABC+CD
 AB(1+C)+CD
 AB+CD

Advantages:
 If we use Boolean algebra for your logical problem
you can save more gates and operations. so your
design will be cheaper, more comprehensible,
more serviceable .
 It allows logical steps quickly and repeatedly.
Disadvantages:
 Can only arrive at direct results not implied once.

boolean algrebra and logic gates in short

boolean algrebra and logic gates in short

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boolean algrebra and logic gates in short