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

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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?
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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.

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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.

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).

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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

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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?

• ? Not

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Can you make a NOT gate from an NAND?
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Can you make an AND gate from an NAND?
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Can you make an OR gate from an NAND?
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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.

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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.

• 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.

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Write out the logic tables for these two circuits.