BOOlean Logic

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BOOlean Logic

• The basis for digital electronics.

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What digital electronics do you use?

• Computer
• CD DVD players
• IPod
• Cell phone
• HDTV
• Digital cameras

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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

4
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.

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Have you used Boolean logic?

• What information have you searched for on the
  Web?
• Do a Google Search for car radios.

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The AND operator (both, all)

• rivers AND salinity
• dairy products AND export AND Europe

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The OR operator (either, any)

• fruit OR vegetables
• fruit OR vegetables OR cereal

from CSIRO Australia
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The NOT operator

• fruit NOT apples
• Look at the Boolean Machine http//kathyschrock.ne
  t/rbs3k/boolean/

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Lets 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.

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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 discusseither stress or anxiety 1127
                 stress AND anxiety Documents that discussboth stress and anxiety 51
                 stress NOT anxiety Documents that discussstress but not anxiety 687
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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.

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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.

13
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.

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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

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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
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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
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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.
• http//www.play-hookey.com/digital/basic_gates.htm
  l

A B Q
0 0 0
0 1 1
1 0 1
1 1 1

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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.

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Construct the logic table for these circuits.
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What happens when you add a NOT to an AND gate?
A B Q
0 0 0
0 1 0
1 0 0
1 1 1

• ? Not

A B Q
0 0 1
0 1 1
1 0 1
1 1 0
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Can you make a NOT gate from an NAND?
A Q
0 1
1 0
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Can you make an AND gate from an NAND?
A B Q
0 0 0
0 1 0
1 0 0
1 1 1
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Can you make an OR gate from an NAND?
A B Q
0 0 0
0 1 1
1 0 1
1 1 1
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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.

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How do we use gates to add two binary numbers?

• Binary numbers are either 1 or 0, either on or
  off.
• Have two outputs.
• Need a gate to produce each output.

0 0 1 1
0 1 0 1
00 01 01 10
A B Q CO
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
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How do I add larger numbers?

• You can write any number in binary.

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Add binary numbers.

• Adding larger number just adds more columns.

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Gang circuits in sequence.

• 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.

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Different Version of Full Adder.
One-bit Full Adder with Carry-In and Carry-Out One-bit Full Adder with Carry-In and Carry-Out One-bit Full Adder with Carry-In and Carry-Out One-bit Full Adder with Carry-In and Carry-Out One-bit Full Adder with Carry-In and Carry-Out

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.

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Add multiple bits.

• 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

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Gates from electrical relays.

• NOT gate
• AND gate
• OR gate

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Test your knowledge on Boolean logic.

• http//educ.queensu.ca/compsci/resources/BoolLogi
  c/assocdistrib.html

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Here is a great resource for digital electronics.

• http//hyperphysics.phy-astr.gsu.edu/Hbase/electro
  nic/digcktcon.htmlc1

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• Following are extra slides which may not be used.

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Binary numbers.
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Exclusive gates

• Exclusively OR gate are true if either input is
  true but not both.

A B Q
0 0 0
0 1 1
1 0 1
1 1 0
A B Q
0 0 1
0 1 0
1 0 0
1 1 1
37
Write out the logic tables for these two circuits.