Activity 2.1.1 AOI Design:
Truth Tables to Logic Expressions
Introduction
The first step in designing a new product is clearly defining the design requirements or design
specifications. These design specifications detail all of the features and limitations of the new
product.
In digital electronics, the process of translating these design specifications into a functioning
circuit starts with the creation of a truth table. A truth table is simply a list of all possible binary
input combinations that could be applied to a circuit and the corresponding binary outputs
that the circuit produces. Once the truth table is complete, a Boolean expression can easily
be written directly from the truth table.
In this activity you will learn how to translate design specifications into truth tables and, in
turn, write un-simplified logic expressions from these truth tables.
In future activities we will learn how to use Boolean algebra as well as a graphical technique
called Karnaugh mapping to simplify these logic expressions.
A truth table shows how a logic design’s output
respond to ALL combinations of possible inputs.
•
A logic design with N inputs will have 2
N
input
combinations.
•
The input are listed in binary order (i.e., counting
order) in the columns to the left.
•
The output(s) are listed in the column(s) to the
right.
(Note some logic circuits can have more
than one output.)
Write the Minterm adjacent to every row in the truth table
that contains a one in the output column.
•
Write the Sum-Of-Products (SOP) logic expression by
summing together all of the Minterms.
For each term in the logic expression, place a one in the
output column for the input condition that matches the
term.
•
Some terms may match more than one input condition.
Identify the number of input variables.
•
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 1
Assign variable names and establish the assignment
condition for each variable (i.e., What does a 0 or 1 mean
for that input?).
•
Create a truth table.
Procedure
Before we tackle the more challenging aspects of translating a written specification into a
truth table, let’s take a look at the process of writing an un-simplified logic expression from a
completed truth table.
Example:
A
B
Z
0
0
0
0
1
1
1
0
1
1
1
0
Minterms
AB
AB
Z  ABAB
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 2
Using the above example (previous page) as a guide, write the un-simplified logic expression for
each of the following truth tables. Though it is not required, you may find it helpful to first write the
Minterm expression for every place containing a one in the output function.
1.
2.
5.
M
N
F1
X
Y
F2
W
X
Y
Z
F5
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
1
1
1
1
1
1
0
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
1
0
0
0
1
1
1
0
1
0
0
0
0
1
0
0
1
0
1
0
1
0
1
1
0
1
1
0
1
1
0
0
1
1
1
0
1
0
1
1
1
0
1
1
1
1
1
0
F1 = M’N’+MN
F2 = X’Y’+XY’+XY
3.
4.
P
Q
R
F3
J
K
L
F4
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
0
1
0
0
0
1
0
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
0
1
1
1
0
0
1
1
1
0
1
1
1
1
F3 =P’Q’R+PQ’R’+PQR’
F4 =J’K’L+J’KL+JK’L’+JKL
F5 =W’X’YZ’+W’XY’Z’+WX’YZ’+WXY’Z’+WXYZ’
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 3
Now that you have mastered the process of writing an un-simplified logic expression from a
completed truth table, let’s reverse the process. For the following logic expressions, create
a corresponding truth table. Note that some terms in the logic expression may map to more
than one place in the truth table.
1.
F1  C D  C D
C
D
F
0
0
1
0
1
1
1
0
0
1
1
0
2.
F2  R S T  R S  R S T  S T
R
S
T
F2
0
0
0
1
0
1
0
1
1
0
1
1
0
1
1
0
1
1
0
0
1
1
1
1
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 4
3.
F3  A B C  A B C  A B C  A C
A
B
C
F3
0
0
0
1
0
1
0
1
1
0
0
1
0
0
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
1
1
1
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 5
Now that you understand the mechanics of converting from a truth table to a logic
expression (and vice-versa), let’s revisit a circuit design we were introduced to in Unit 1.
Your new car has an audio alarm that buzzes whenever the door is open and the key is in
the ignition or when the key is in the ignition and the seatbelt is not buckled.
NOTE: This seatbelt alarm design is slightly different than the one you created
earlier.
1. Using the following variable names and assignment conditions:
Complete the truth table shown below that captures the functionality of this
audio alarm.

D: Door → 0=Door Open / 1=Door Close

K: Key → 0=Key Not in Ignition / 1=Key in Ignition

S: Seat → 0=Not Buckled / 1=Buckled

B: Buzzer → 0=Buzzer Off / 1=Buzzer On
D
K
S
B
(Door)
(Key)
(Seat Belt)
(Buzzer)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
0
2. Using the truth table you just created, write the un-simplified logic expression for
the buzzer (i.e., variable B). Be sure that your answer is in the Sum-of-Products
form.
B= D’KS’+D’KS+DKS’
In a future activity, you will see that this logic expression can be simplified to:
B  DK K S
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 6
Note: Some of you may have been able to extract this simplified logic expression directly from the
specifications. If so, you may be asking yourself, why did I have to go through the process of
creating the truth table and writing the un-simplified logic expression first? This was a simple
problem. As you progress through this class, the design problems will become more difficult The
process that you are learning now will become invaluable in solving more challenging problems.
Your Aunt Mary would like you to design a digital logic circuit that
monitors the conditions in a room of her house where her seven
cats live. The room’s temperature is monitored by a temperature
sensor. The room’s humidity is monitored by a humidity sensor. The
air quality of the room is monitored by a special sensor that
measures air particle density. The three sensors output a one (1) to
indicate an out-of-range condition. Located outside of the room is an
ALERT light that is on (1) whenever two or more sensors are out-ofrange.
1. Create a truth table that captures the functionality of this room monitoring system.
T
H
A
L
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
1
1
0
1
0
1
1
1
1
0
1
1
1
1
1
1
2. Using the truth table that you just created, write the un-simplified logic expression
for the ALERT light. Be sure that your answer is in the Sum-of-Products form.
L= THA’+T’HA+TH’A+THA
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 7
Conclusion
A digital logic circuit with (2) inputs has (4) input combinations. One with (3) inputs has (8)
combinations. One with (4) inputs has (16) combinations.
1. How many input combinations would a digital logic circuit have if it has (5) inputs?
How about (6) inputs?
5 inputs = 32 combinations
6 inputs= 64 combinations
2. Mathematically express the relationship between the number of input (N) and the
number of input combinations (C).
2n=C
3. Write the un-simplified logic expression for the truth table shown below.
X
Y
Z
0
0
1
0
1
1
1
0
1
1
1
1
Z= X’Y’+X’Y+XY’+XY
Did you see the short cut? (Hint: Z is always a one.)
Going Further – Optional
The un-simplified logic expression for the truth table shown below is;
C  ABABAB
A
B
C
Another way to write this would be C 
0
0
1
Explain why this is true.
0
1
1
1
0
0
1
1
1
A B.
This would be true because C = 0 when A=1 and B=0.
© 2014 Project Lead The Way, Inc.
Digital Electronics Activity 2.1.1 AOI Design: Truth Tables to Logic Expressions – Page 8