Download Chapter 3.5

Chapter 3.5
Logic Circuits
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.
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”.
“And” Gate

Wires labeled a and b contain an “input”
voltage that either represents “1” or “0”. The
“output” voltage, labeled a b is given by this
“truth table”:
a
0
0
1
1
b
0
1
0
1
a b
0
0
0
1
“Or” Gate

Wires labeled a and b contain an “input”
voltage that either represents “1” or “0”. The
“output” voltage, labeled a  b is given by this
“truth table”:
a
b
a+b
0
0
0
0
1
1
1
0
1
1
1
1
“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
1
a’
0
0
1
Building a logic circuit

Using the “and”, “or”, and “inverter” gates,
we can design more complicated circuits.
Consider the following circuit. What outputs will be
obtained for different combinations of input?
a
b
1
1
1
0
0
1
0
0
How many gates are there?
In the previous example there was a twoinput or gate, a two-input and gate, and a
not gate.
 Is there an equivalent circuit which uses
less gates?

Write the Boolean algebra expression which
corresponds to the following circuit:
Use the laws of Boolean algebra to
simplify the last expression.

How many gates can be saved?
Write the Boolean algebra expression which
corresponds to the following circuit:
Use the laws of Boolean algebra to
simplify the last expression.

How many gates can be saved?
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.
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.
Simplify x’y + x’y’ + xy using a Karnaugh map
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.
Simplify x’yz + x’yz’ + xyz’+ x’y’z using a
Karnaugh map
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?
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?
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.)
Simplify xy’z’ + x’z + xy using a Karnaugh
map