digital logic gates and boolean algebra
DESCRIPTION
Useful for B.Sc III year StudentsTRANSCRIPT
![Page 1: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/1.jpg)
DIGITAL LOGIC GATES AND BOOLEAN ALGEBRA
Dr. C. SARITHA
LECTURER IN ELECTRONICS
SSBN DEGREE & PG COLLEGE
ANANTAPUR
![Page 2: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/2.jpg)
LOGIC GATES
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.
![Page 3: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/3.jpg)
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 with the advent of integrate circuits, switches have been replaced by TTL circuit and CMOS circuits.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.
![Page 4: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/4.jpg)
TYPES OF LOGIC GATES:The most common logic gates used are,Basic gates 1.OR2.AND3.NOTUniversal gates1.NAND2.NOR X-OR or Exclusive-OR
![Page 5: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/5.jpg)
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:
![Page 6: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/6.jpg)
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.
![Page 7: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/7.jpg)
TRUTH TABLE:
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
![Page 8: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/8.jpg)
AND GATE: The AND gate is also a basic kind of
digital circuit. It has also two or more inputs and one
output.
SYMBOL:
![Page 9: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/9.jpg)
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.
![Page 10: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/10.jpg)
TRUTH TABLE:
A B A+B
0 0 0
0 1 0
1 0 0
1 1 1
![Page 11: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/11.jpg)
NOT GATE or Inverter Gate: A NOT gate is a basic gate that has one
input and one output.
SYMBOL:
![Page 12: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/12.jpg)
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
![Page 13: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/13.jpg)
TRUTH TABLE:
A Ā
0 1
1 0
![Page 14: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/14.jpg)
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:
![Page 15: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/15.jpg)
The Boolean expression for the NAND operation is given by,
Y=A.B
![Page 16: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/16.jpg)
TRUTH TABLE:
A B AB
0 0 1
0 1 1
1 0 1
1 1 0
![Page 17: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/17.jpg)
NOR GATE: The NOR gate is also a universal gate
and it is a combination of a NOT and OR gates.
SYMBOL:
![Page 18: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/18.jpg)
The Boolean expression for NOR gate is given by,
Y=A+B
![Page 19: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/19.jpg)
TRUTH TABLE:
A B A+B
0 0 1
0 1 0
1 0 0
1 1 0
![Page 20: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/20.jpg)
Exclusive OR or X-OR GATE: The X-OR gate is a logic gate having two
inputs with and single output.
SYMBOL:
![Page 21: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/21.jpg)
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
+
+
![Page 22: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/22.jpg)
TRUTH TABLE:
A B AB+AB
0 0 0
0 1 1
1 0 1
1 1 0
![Page 23: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/23.jpg)
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.
![Page 24: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/24.jpg)
BOOLEAN ALGEBRA
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.
![Page 25: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/25.jpg)
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
![Page 26: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/26.jpg)
AND laws: A.0=0 A.A=A A.1=A A.Ā=0NOT laws: 0=1 1=0 If A=0 then Ā=1 If A=1 then Ā=0
Ā=A
![Page 27: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/27.jpg)
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
![Page 28: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/28.jpg)
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)=ABDemorgan’s laws: A+B=A.B A.B=A+B
![Page 29: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/29.jpg)
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
![Page 30: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/30.jpg)
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.
![Page 31: Digital logic gates and Boolean algebra](https://reader035.vdocument.in/reader035/viewer/2022081413/548b6fbab4795940568b4675/html5/thumbnails/31.jpg)