Logic Circuits

1
Chapter 3.5

• Logic Circuits

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How does Boolean algebra relate to computer
circuits?

• Data is stored and manipulated in a computer as a
  binary number.
• Individual bits of the number are represented
  with two different voltage levels, 0 and 1.
• Bits are combined using complicated circuits to
  do operations such as integer arithmetic.

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Example Add 75 and 3

• Given a string, 0000000001001011 and a string
  0000000000000011 it creates the string
  0000000001001110.
• This is accomplished using simple circuits called
  gates.

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And Gate

• Wires labeled a and b contain an input voltage
  that either represents 1 or 0. The output
  voltage, labeled is given by this truth
  table

a b
0 0 0
0 1 0
1 0 0
1 1 1
5
Or Gate

• Wires labeled a and b contain an input voltage
  that either represents 1 or 0. The output
  voltage, labeled is given by this truth
  table

a b ab
0 0 0
0 1 1
1 0 1
1 1 1
6
Inverter Gate

• A wire labeled a contains an input voltage that
  either represents 1 or 0. The output
  voltage, labeled a is given by this truth
  table

a a
1 0
0 1
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Building a logic circuit

• Using the and, or, and inverter gates, we
  can design more complicated circuits.

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Consider the following circuit. What outputs
will be obtained for different combinations of
input?
a b
1 1
1 0
0 1
0 0
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How many gates are there?

• In the previous example there was a two-input or
  gate, a two-input and gate, and a not gate.
• Is there an equivalent circuit which uses less
  gates?

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Write the Boolean algebra expression which
corresponds to the following circuit
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Use the laws of Boolean algebra to simplify the
last expression.

• How many gates can be saved?

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Write the Boolean algebra expression which
corresponds to the following circuit
13
Use the laws of Boolean algebra to simplify the
last expression.

• How many gates can be saved?

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Sums of Products

• Two examples of sums of products are xyyx
  and xyz xyz xyz
• Karnaugh maps is a useful graphical technique for
  simplifying Boolean algebra expressions such as
  these and they give the simplest possible
  sums-of-products expression.

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Simplify xy xy using a Karnaugh map

• Check the boxes that correspond to xy and xy.
• Circle any rectangle shapes formed by the checks.
• Determine the variable that will not appear in
  the simplified answer.

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Simplify xy xy xy using a Karnaugh map
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Karnaugh maps for 3 variables

• Use the map shown.
• Along the top, labels that are side by side
  differ in exactly one of the two variables.
• Check the appropriate boxes.
• Note 1x1 squares do not remove any variables a
  vertical or horizontal circle of area 2 removes
  one variable.

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Simplify xyz xyz xyz xyz using a
Karnaugh map
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Simplify xyz xyz xyz xyz xyz using
a Karnaugh map

• What is the simplified expression?
• Is yzyzxy the simplest expression?

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Simplify xyz xyz xyz xyz xyz using
a Karnaugh map

• Note yzyzxy is NOT the simplest expression.
• What is the simplified expression?

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Guidelines for choosing rectangles

• Choose rectangles so that the number of
  rectangles is as small as possible and each
  individual rectangle is as large as possible (but
  sides of length 3 are not allowed.)

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Simplify xyz xz xy using a Karnaugh map