The document provides an overview of Boolean algebra, which is essential for simplifying logic expressions used in digital circuits. It covers the basic definitions, operations, and theorems of Boolean algebra and explains its practical applications, including the functioning of logic gates in everyday technology. Additionally, it emphasizes the importance of using Boolean algebra for cost-effective circuit design in modern computing.

1. Demo Class
Boolean Algebra
by:
Hau Fung Moy Kwan, Ph.D
June, 18 2018
Education is not the learning of facts, but the training of the mind to think- ALBERT EINSTEIN

2. Table of contents
 Introduction
 Basic definition
 Basic operations of boolean algebra
 Boolean theorems
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3. Introduction
 Because binary logic is used in all of
today´s digital computers and devices,
the cost of the circuit that implement
it is important factor addressed by
designers- be they computer
engineers, electrical engineers, or
computer scientist.
 Finding simpler and cheaper, but
equivalent, realizations of a circuit
can reap huge payoffs in reducing the
overall cost of the design.
 Mathematical methods that simplify
circuits rely primarily on Boolean
algebra.
Hau Fung Moy Kwan, PhD.
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4. What is Boolean algebra?
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5. Where do you think Boolean algebra is used now
a days?
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6. Boolean algebra
Basic Definition
 Boolean algebra; it is the set of rules used to simplify by
given logic expression without changing its functionality.
 Boolean algebra is a logical calculus of truth values. It deals
with two-values (true / false or 1 and 0) variables. The
symbols 0 and 1 are called bits.
 Bit: a bit (short for binary digit) is the smallest unit
of data in a computer.
Hau Fung Moy Kwan, PhD.
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7. Basic operations of Boolean algebra
In Boolean algebra there are only three basic
operations:
1.- Logical Addition; OR Addition; OR operation.
The common symbol is the plus sign (+).
2.-Logical multiplication; AND multiplication; AND
operation.
The common symbol is the multiplication sign (.)
3.- Logical complementation or inversion. Also called the
NOT operation( )
Hau Fung Moy Kwan, PhD.
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8. Basic operations of Boolean algebra
The three basic logical operations are:
1.- OR
2.- AND
3.-NOT
OR operation:
 If we have two independents variable A and B and we want to see what
the the result is when all together.
X= A + B
 We can define the rules OR sum in a table. A and B are variable that
represent logic variable. A and B can be either 0 or 1 two possible values
Hau Fung Moy Kwan, PhD.
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9. Basic operations of Boolean algebra
AND operation:
 If we have two varaibles two logic variables A and B and we want to know
what the result is X = A . B
 We can use the truth table to define the rules for the AND operation
 Let’s build it:
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10. Basic operations of Boolean algebra
Not Operation:
 The NOT operation is unlike the or and AND operation in that it can be
performed on a single variable.
 So, if we have a variable A subjective with a NOT operation the result X
will be express:
X= A Read as X equals Not A
We can summarize this rules in a truth table with:
Hau Fung Moy Kwan, PhD.
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11. A B X=A + B
0 0 0
0 1 1
1 0 1
1 1 1
Basic operations of Boolean algebra
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OR operation:
Truth table with A and B as variables
X= A +B
Symbol OR operation
A B C X=A + B
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
OR operation:
Truth table with A , B and C as variables
X= A +B + C
A
symbol OR operation
B
C

12. Basic operations of Boolean algebra
A B X=A . B
0 0 0
0 1 0
1 0 0
1 1 1
Truth table with A and B as variables Symbol AND operation
A
B
X= A . B
Note: like ordinary multiplication with AND multiplication you do not include
the dot is understood. Keep it out or leave it out. X=AB
A X= A
0 1
1 0
Symbol NOT operation
A X=A
inverter
Complementation, Inversion
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13. Basic operations of Boolean algebra
OR
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
Let’s summarize:
AND
0 . 0 = 0
0 . 1 = 0
1 . 0 = 0
1 . 1 = 1
NOT
0 = 1
1 = 0
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14. Boolean Theorems
 Boolean algebra can be used to analyzed a logic circuit and
express it is operation mathematically.
 Once the Boolean expresion for a logic circuit has been
obtained it may often be reduced to a simpler form using
boolean algebra
 The most important application of Boolean algebra that is the
simplification of logic circuits.
 By using Boolean theorems to simplify logic expressions
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15.  The first group of Boolean theorems we are going to define
are:
1. Single variable theorems: we are going to dealing with one
variable.
X, X=0 or X=1
Boolean Theorems
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16. 2.- Multivariable theorems:
Commutative laws;
X+ Y= Y+X
X.Y= Y.X
The order in which we add or multiply two variables in
unimportant.
Associative law;
X +(Y+Z)= (X+Y)+Z = X+Y+Z
X(Y.Z)= (X.Y).Z = XYZ
We can group the terms of a sum or product any way we want.
Boolean Theorems
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17. Boolean Theorems
2.- Multivariable theorems:
Distributive law;
X(Y+Z)= XY + XZ
An expression can be expanded by multiplying term by term
X+XY =X
X +XY=X+Y
Now, let’s simplify an expression Y=ABD+ABD
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18. X
X
X
1
X
0
X
X.0 = 0
X.1 = X
X.X = X
X. X = 0
X+0 = X
X + 1= 1
X+X = X
X+X = 1
X
0
0
X
X
1
Boolean Theorems
Single variable theorems
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X X
X
0
X
X
X
0
1
X
1
Let’s summarize:

19. Boolean Theorems
Multivariable theorems
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Commutative laws;
X+ Y= Y+X
X.Y= Y.X
Associative law;
X +(Y+Z)= (X+Y)+Z = X+Y+Z
X(Y.Z)= (X.Y).Z = XYZ
Distributive law;
X(Y+Z)= XY + XZ
X+XY =X
X +XY=X+Y
Let’s summarize:

20. Let’s proof some Boolean theorems
1. x + x = x
2. DeMorgan’s theorem, 𝑥 + 𝑦 = 𝑥 𝑦, their validity is easily shown with truth tables.
For example:
X Y X+ Y ( X + Y )
0 0 0 1
0 1 1 0
1 0 1 0
1 1 1 0
X Y ( X + Y )
1 1 1
1 0 0
0 1 0
0 0 0
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21. What are the practical applications of logic
gates?
 You practically applied Millions of logic gates to inquire as to what is the practical
application of logic gates.
 That's because the logic gates are the building blocks for all computers, smart
phones and the whole internet. Let’s see some day to day applications of the 3 basic
gates, that a common person may relate to:
OR Gate
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ON/OFF
Front Doorbell Switch
Back Doorbell Switch
0 Doorbell
ON/OFF
If either the Front Door switch OR the
Back Doorbell Switch is pressedthen
the Doorbell rings

22. AND Gate
What are the practical applications of logic
gates?
ON/OFF
Person Sensor
Alarm Switch
0
Burglar Alarm
ON/OFF
ON/OFF
If both the Person Sensor AND the
Alarm Switch are on then the Burglar
Alarm is activated
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23. What are the practical applications of logic
gates?
NOT Gate
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Tempeture Detector
(Above 20 ℃ )
ON/OFF
ON/OFF
If the Temperature is Above 20 ℃ then
the Central heating is switch off.
If the Temperature is below20 ℃ then
the Central heating is switch on.
Central Heating

24. Bibliography
Books:
Online web pages to practices:
http://www.ee.surrey.ac.uk/Projects/Labview/boolalgebra/quiz/index.html
https://math.stackexchange.com/questions/653201/exercise-regarding-boolean-algebra
Personal Blog: https://sites.google.com/site/haumoy/ Canvas
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25. Thank You!!!
“A person who never made a mistake never tried anything new”
“Never give up on what you really want to do.
The person with big dreams is more powerful than one with all the facts”
-Albert Eistein
amuicita@yahoo.com 385-202-9927
https://www.linkedin.com/in/hau-moy-4b491620/
https://sites.google.com/site/haumoy/ Canvas
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