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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”:
ba
a b
0 0 0
0 1 0
1 0 0
1 1 1
ba
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“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”:
ba
a b a+b
0 0 0
0 1 1
1 0 1
1 1 1
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“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:
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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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Sums of Products
Two examples of sums of products are xy’+yx’ and xy’z + x’y’z + x’y’z’
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’ + x’y’ using a Karnaugh map
Check the boxes that correspond to xy’ and x’y’.
Circle any rectangle shapes formed by the checks.
Determine the variable that will not appear in the simplified answer.
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Simplify x’y + x’y’ + 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 x’yz + x’yz’ + xyz’+ x’y’z using a Karnaugh map
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Simplify x’y’z + x’yz’ + x’yz + xy’z + xyz using a Karnaugh map
What is the simplified expression?
Is yz+y’z+x’y the simplest expression?
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Simplify x’y’z + x’yz’ + x’yz + xy’z + xyz using a Karnaugh map
Note: yz+y’z+x’y 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 xy’z’ + x’z + xy using a Karnaugh map