Recall '''Numbers in Binary

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Recall ...Numbers in Binary

• Remember
• Binary representations of...
• Symbols
• Images
• Music
• In this lecture
• Further binary representations of data
• Sound
• Instructions
• Binary logic

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Representation of Music
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Representation of Sound
sound waves
paper cone
magnet
voltage wires
coil
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Representation of Sound
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Representing anything in Binary

• Programs/Instructions?
• A program is a sequence of tasks
• each unique task can be represented by a sequence
  of bits
• seems nearly anything can be represented in
  binary...
• So, how did we wind up with bits anyway?

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Logic and Logic Circuits

• What is a circuit?
• the complete path of an electric current, or a
  collection of electronic elements
• We are interested in logic circuits, those whose
  output varies depending on their input
• Logic circuits emulate the Boolean logic
  operators from propositional logic

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Binary for Logic?

• In Propositional Logic we examine the truth of
  logical sentences across all possible scenarios.
• Sentences consist of
• Variables
• sub-sentences (A, B, C ) which are either true
  or false
• In computer science, we use
• true 1
• false 0
• Logical Operators
• e.g., NOT, OR, AND, XOR, NAND, NOR
• Example sentence
• X (A AND B) OR C
• We examine the logical outcome of a sentence
  through truth table analysis

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Simplest Truth Tablea Single Variable
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Logical AND
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Logical AND on two variables

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Logical Or

• A OR B
• A ? B
• Has two values true if either A or B is true, or
  if both A and B are true
• false if they are both false.
• Are either of these things true?
• Note both can be true this is not exclusive or

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Logical Not

• Used to invert a meaning
• NOT A as an alternative
• A It is raining
• ?A It is not raining

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XOR- Exclusive Or

• True when either A or B are true, but not both
• So A and NOT B or B and NOT A
• Built from simpler logic
• (A ? ?B) ? (?A ? B)

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Nor and Nand
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Truth Table Analysis

• How do you build a truth table?
• Step 1 Create columns for all the variables in
  the sentence
• Step 2 Determine the number of rows you need
  given the variables in your sentence
• Step 3 Define all possible sequences (cases) for
  your truth table, starting with all variables
  false and ending with all variables true
• Step 4 De-construct the logic in the sentence
  and fill in your table
• What is the truth table for
• X A AND (NOT B)
• X (NOT A) AND (NOT B)
• X (A OR B) AND C

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Logic and Logic Circuits

• What is a circuit?
• the complete path of an electric current, or a
  collection of electronic elements
• we will consider transistors to be the basic
  building blocks of logic computer hardware.
• Logic circuits are built from a series of
  transistors
• What is a transistor?

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Transistors

• A transistor is an electronic device that has
  three ends a source, a sink, and a gate
• In this type of transistor, when the gate is
• ON, power flows from the source to the sink.
• OFF, power does not flow to the sink

source
gate
sink
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Transistors (like faucets)

• The operation of a transistor could be explained
  by making an analogy to faucets.
• A faucet has
• An input
• the water company
• source
• An output
• a sink (where water is drained)
• sink
• Flow control
• If the tap knob (gate) is turned
• ON water flows from the source to the sink
• OFF no water flows.
• The state of the tap determines the presence of
  water at the sink

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Transistors (like faucets)

• Changing from water to electricity in
  transistors
• Electricity flows from the source to the sink
  with the gate 1 (ON)
• Electricity does not flow from the source to the
  sink with the gate 0 (OFF)

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Transistors

• The current technology used to build computer
  hardware (chips) is called CMOS.
• In CMOS we also use another kind of transistor,
  distinguished by the little bubble
• The bubble means that this transistor works in
  the opposite way (it's ON when the gate is OFF
  and OFF when the gate is ON).

source
gate
sink
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Building Complicated Circuits with Transistors
Battery

• What on earth does this do?
• If A0 then . . .
• bottom gate is off
• top gate is on
• power flows from the battery, cant go out the
  sink, and goes out through Z. Z 1
• If A 1 then
• bottom gate is on
• top gate is off
• power doesnt get to Z from the battery, and any
  power left in Z will flow out the sink. Z 0

A
Z
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Building Circuits with Transistors
Battery
A
Z
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Building Circuits with Transistors
Battery
What does this circuit do? -- build a truth
table!
A
B
Z
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Building Circuits with Transistors
Battery
A
B
Z
Looks like a negated AND (NAND) Attach a NOT
circuit to the output of Z and wed get AND
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Building Circuits with Transistors
Battery
A
B
Z
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Logic Gates