Transcript BOOlean Logic - Johns Hopkins University

BOOlean Logic
The basis for digital electronics.
What digital electronics do you
use?
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Computer
CD & DVD players
IPod
Cell phone
HDTV
Digital cameras
What are digital electronics?
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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.
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DVD-Audio can be 96 or 192 kHz
and up to 24 bits resolution
George Boole
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George Boole, a British mathematician (1815-1864).
 Logic and math are equivalent.
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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?
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What information have you searched for on the Web?
Do a Google Search for car radios.
The AND operator
(both, all)
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rivers AND salinity
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dairy products AND export AND
Europe
The OR operator
(either, any)
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fruit OR vegetables
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fruit OR vegetables OR cereal
from CSIRO Australia
The NOT operator
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fruit NOT apples
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Look at the Boolean Machine
http://kathyschrock.net/rbs3k/boolean/
Let’s use Boolean logic to
examine class.
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Please stand up if you are:
 girl
 AND black hair
 AND left handed
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Please stand up if you are:
 girl
 OR black hair
 OR left handed
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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
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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
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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
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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.
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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.
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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
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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
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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
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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.
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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?
 Not
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.
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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?
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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?
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You can write any number in binary.
Add binary numbers.
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Adding larger number just adds more columns.
Gang circuits in sequence.
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To construct a full adder circuit, we'll
need to gang these simple circuits
together.
The “carry” value will become an input
for the subsequent
Now have three inputs and two outputs.
Since we'll have both an input carry and
an output carry, we'll designate them as
CIN and COUT.
Different Version of Full Adder.
One-bit Full Adder with Carry-In and Carry-Out
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CI
A
B
Q
CO
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
Now we can add two binary bits together
 accounting for a possible carry from the next lower order of magnitude.
 sending a carry to the next higher order of magnitude.
Add multiple bits.
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A full adder must be allocated for each bit to be added
simultaneously.
To add two 4-bit numbers to produce a 4-bit sum (with a
possible carry), you would need four full adders with carry
lines cascaded.
By extension, two binary numbers of any size may be added in
this manner.
http://www.play-hookey.com/digital/adder.html
Gates from electrical relays.
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NOT gate
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AND gate
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OR gate
Test your knowledge on
Boolean logic.
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http://educ.queensu.ca/~compsci/resources/BoolLogic
/assocdistrib.html
Here is a great resource for
digital electronics.
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http://hyperphysics.phyastr.gsu.edu/Hbase/electronic/digcktcon.html#c1
Following are extra slides which may not be used.
Binary numbers.
“Exclusive” gates
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Exclusively OR gate are true if either input is true but
not both.
A
B
Q
A
B
Q
0
0
0
0
0
1
0
1
1
0
1
0
1
0
1
1
0
0
1
1
0
1
1
1
Write out the logic tables for
these two circuits.