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 is given by this “truth table”:

ba

a b

0 0 0

0 1 0

1 0 0

1 1 1

ba

“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

“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

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 two-input 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