Computer Science 101 Survey of Computer Science

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Computer Science 101Survey of Computer Science

• Introduction to
• Logic Circuits

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From Boolean Logic to Hardware

• Boolean values true and false map to binary
  values 1 and 0
• Binary values 1 and 0 map to discrete states on
  and off
• Discrete states on and off map to voltage levels
  5V and 0V on an analog device
• Connect these devices to perform logical
  operations

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Transistor - Electronic Switch

• Base High (5v or 1) Makes connection
• Base Low (0v or 0) Disconnects

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AND Gate
5v

• Output is 1 only if
• Input-1 is 1 and
• Input-2 is 1
• Output Input1 AND Input2

Output
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OR Gate

• Output is 1 if
• A is 1 or if
• B is 1
• Output A OR B

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NOT Gate

• Input High (5v or 1) Output Low (0v or 0)
• Input Low (0v or 0) Output High (5v or 1)
• Output is opposite of Input

5v
Output
Input
Ground
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Boolean Exp ? Logic Circuit

• To draw a circuit from a Boolean expression
• From the left, make an input line for each
  variable.
• Next, put a NOT gate in for each variable that
  appears negated in the expression.
• Still working from left to right, build up
  circuits for the subexpressions, from simple to
  complex.

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Logic Circuit _ ____ AB(AB)B
Input Lines for Variables
A
B
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Logic Circuit _ ____ AB(AB)B
NOT Gate for B
A
B
_ B
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Logic Circuit _ ____ AB(AB)B
_ Subexpression AB
_ AB
A
B
_ B
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Logic Circuit _ ____ AB(AB)B
Subexpression AB
_ AB
A
AB
B
_ B
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Logic Circuit _ ____ AB(AB)B
___ Subexpression AB
_ AB
A
____ AB
AB
B
_ B
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Logic Circuit _ ____ AB(AB)B
___ Subexpression (AB)B
_ AB
A
____ AB
AB
B
_ B
____ (AB)B
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Logic Circuit _ ____ AB(AB)B
Entire Expression
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Logic Circuit ? Boolean Exp

• In the opposite direction, given a logic circuit,
  we can write a Boolean expression for the
  circuit.
• First we label each input line as a variable.
• Then we move from the inputs labeling the outputs
  from the gates.
• As soon as the input lines to a gate are labeled,
  we can label the output line.
• The label on the circuit output is the result.

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Logic Circuit ? Boolean Exp
_ _ ABAB
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Simplifying Boolean Expressions

• As in ordinary algebra, some Boolean expressions
  can be simplified
• The logic circuit that results from a simplified
  expression will have fewer gates and operations

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Laws of Boolean Algebra
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Laws of Boolean Algebra
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Simplification Revisited

• Once we have the BE for the circuit, perhaps we
  can simplify.

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Logic Circuit ? Boolean Exp
Reduces to
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The Boolean Triangle
Boolean Expression
Logic Circuit
Truth Table
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The Boolean Triangle
Boolean Expression
Logic Circuit
Truth Table