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.

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
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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
Hau Fung Moy Kwan, PhD.
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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
Hau Fung Moy Kwan, PhD.
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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
Hau Fung Moy Kwan, PhD.
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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
Hau Fung Moy Kwan, PhD.
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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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