BIRLA VISHVAKARMA MAHAVDYALAYA V. V. NAGAR
BOOLEAN ALGEBRA
Guided By: Prof. P. B. Swadas
Prepared By:
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What digital electronics do
you use?
Computer
CD & DVD players
IPod
Cell phone
HDTV
Digital cameras
What are digital electronics?
Sound is an analog signal.
On a CD, digital sound is encoded as 44.1
kHz, 16 bit audio.
◦ The original wave is 'sliced' 44,100 times a
second - and an average amplitude level is
applied to each sample.
◦ 16 bit means that a total of 65,536 different
values can be assigned, or quantized to
each sample.
DVD-Audio can be 96 or 192 kHz and up to
24 bits resolution
George Boole
George Boole, a British mathematician (1815-1864).
◦ Logic and math are equivalent.
All math functions can be determined using these 3 primary Boolean
logic operators: AND, OR, and NOT.
◦ AND narrows your search,
◦ OR broadens your search, and
◦ NOT is used to exclude concepts.
Have you used Boolean logic?
What information have you searched for on the Web?
Do a Google Search for car radios.
The AND operator
(both, all)
rivers AND salinity
dairy products AND export AND Europe
The OR operator
(either, any)
fruit OR vegetables
fruit OR vegetables OR cereal
from CSIRO Australia
The NOT operator
fruit NOT apples
Look at the Boolean Machine http://kathyschrock.net/rbs3k/boolean/
Let’s use Boolean logic to
examine class.
Please stand up if you are:
◦ girl
◦ AND black hair
◦ AND left handed
Please stand up if you are:
◦ girl
◦ OR black hair
◦ OR left handed
And NOT
How has the group changed depending on the logical operator used.
What would a library search give
for “stress” and “anxiety”?
Shaded areas indicate
retrieval
Search Statement
Types of records retrieved
Number of
records
retrieved
stress OR anxiety
Documents that discuss
either stress or anxiety
1127
stress AND anxiety
Documents that discuss
both stress and anxiety
51
stress NOT anxiety
Documents that discuss
stress but not anxiety
687
Nesting
When more than one element is in parentheses, the sequence is left to
right. This is called "nesting."
◦ (foxes OR rabbits) AND pest control
◦ foxes OR rabbits AND pest control
◦ (animal pests OR pest animals) NOT rabbits
Parentheses should be used to group terms joined by OR when there is
any other operator in the search.
Order of precedence of
Boolean operators
The order of operations is: AND, NOT, OR, XOR
Parentheses are used to override priority.
Expressions in parentheses are processed first.
Parentheses are used to organize the sequence and groups of concepts.
Write out logic statements using
Boolean operators for these.
You have a buzzer in your car that sounds when your keys are in the
ignition and the door is open.
You have a fire alarm installed in your house. This alarm will sound if it
senses heat or smoke.
There is an election coming up. People are allowed to vote if they are a
citizen and they are 18.
To complete an assignment the students must do a presentation or
write an essay.
Basis for digital computers.
The true-false nature of Boolean logic
makes it compatible with binary logic
used in digital computers.
Electronic circuits can produce Boolean
logic operations.
Circuits are called gates.
◦ NOT
◦ AND
◦ OR
NOT gate
The simplest possible gate is called an "inverter," or a NOT gate.
One bit as input produces its opposite as output.
The symbol for a NOT gate in circuit diagrams is shown below.
The logic table for the NOT gate shows input and output.
A
Q
0
1
1
0
From How Stuff Works
AND gate
The AND gate has the following symbol and logic table.
Two or more input bits produce one output bit.
Both inputs must be true (1) for the output to be true.
Otherwise the output is false (0).
A
B
Q
0
0
0
0
1
0
1
0
0
1
1
1
OR gate
The OR gate has the following symbol and logic table.
Two or more input bits produce one output bit.
Either inputs must be true (1) for the output to be true.
A
B
Q
0
0
0
0
1
1
1
0
1
1
1
1
http://www.play-hookey.com/digital/basic_gates.html
Combine gates.
Gates can be combined.
The output of one gate can become the input of another.
Try to determine the logic table for this circuit.
Construct the logic table for
these circuits.
What happens when you
add a NOT to an AND gate?

N
ot
A
B
Q
0
0
1
0
1
1
1
0
1
1
1
0
A
B
Q
0
0
0
0
1
0
1
0
0
1
1
1
Can you make a NOT gate
from an NAND?
A
Q
0
1
1
0
Can you make an AND gate
from an NAND?
A
B
Q
0
0
0
0
1
0
1
0
0
1
1
1
Can you make an OR gate
from an NAND?
A
B
Q
0
0
0
0
1
1
1
0
1
1
1
1
All computations can be
done with an NAND gate.
The most sophisticated computer can
be made with NAND gates from Radio
Shack.
Need millions of them.
How do we use gates to add
two binary numbers?
Binary numbers are either 1 or 0, either on or off.
0
0
1
1
+0
+1
+0
+1
00
01
01
10
Have two outputs.
Need a gate to produce each output.
A
B
Q
CO
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
How do I add larger numbers?
You can write any number in binary.
Add binary numbers.
Adding larger number just adds more columns.