Module 13—Introduction to Number Systems and Logic
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, 1-1 to 1-10
1-11 to 1-20
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FUNDAMENTAL LOGIC CIRCUITS
Upon completing this chapter, you should be able to do the following:
general logic conditions, logic states, logic levels, and positive and negative logic as these terms and
characteristics apply to the inputs and outputs of fundamental logic circuits.
Identify the following logic circuit gates and interpret and solve the associated Truth Tables:
c. Inverters (NOT circuits)
3. Identify variations of the fundamental logic gates and interpret
the associated Truth Tables.
4. Determine the output expressions of logic gates in
5. Recognize the laws, theorems, and purposes of Boolean algebra.
In chapter 1 you learned that the two digits of the binary number system can be represented by the state or
condition of electrical or electronic devices. A binary 1 can be represented by a switch that is closed, a lamp
that is lit, or a transistor that is conducting. Conversely, a binary 0 would be represented by the same devices
in the opposite state: the switch open, the lamp off, or the transistor in cut-off.
In this chapter you
will study the four basic logic gates that make up the foundation for digital equipment. You will see the types of
logic that are used in equipment to accomplish the desired results. This chapter includes an introduction to
Boolean algebra, the logic mathematics system used with digital equipment. Certain Boolean expressions are used in
explanation of the basic logic gates, and their expressions will be used as each logic gate is introduced.
Logic is defined as the science of reasoning. In other words, it is the development of a reasonable or logical
conclusion based on known information.
Consider the following example: If it is true that all Navy
ships are gray and the USS Lincoln is a Navy ship, then you would reach the logical conclusion that the USS
Lincoln is gray.
To reach a logical conclusion, you must assume the qualifying statement is a condition of
truth. For each statement there is also a corresponding false condition. The statement "USS Lincoln is a Navy
ship" is true; therefore, the statement "USS Lincoln is not a Navy ship" is false. There are no in-between
Computers operate on the principle of logic and use the TRUE and FALSE logic
conditions of a logical statement to make a programmed decision.
The conditions of a statement can be
represented by symbols (variables); for instance, the statement "Today is payday" might be represented by the
symbol P. If today actually is payday, then P is TRUE. If today is not payday, then P is FALSE. As you can see, a
statement has two conditions. In computers, these two conditions are represented by electronic circuits operating
in two LOGIC STATES. These logic states are 0 (zero) and 1 (one). Respectively, 0 and 1 represent the FALSE and
TRUE conditions of a statement.
When the TRUE and FALSE conditions are converted to electrical signals,
they are referred to as LOGIC LEVELS called HIGH and LOW. The 1 state might be represented by the
presence of an electrical signal (HIGH), while the 0 state might be represented by the absence of an electrical
If the statement "Today is payday" is FALSE, then the statement "Today is NOT payday" must
be TRUE. This is called the COMPLEMENT of the original statement. In the case of computer math, complement is
defined as the opposite or negative form of the original statement or variable. If today were payday, then the
statement "Today is not payday" would be FALSE. The complement is shown by placing a bar, or VINCULUM, over the
statement symbol (in this case, P ). This variable is spoken as NOT P. Table 2-1 shows this concept and the
relationship with logic states and logic levels.
Table 2-1.—Relationship of Digital Logic Concepts and Terms
In some cases, more than one variable is used in a single expression. For example, the expression AB
C D is spoken "A AND B AND NOT C AND D."
POSITIVE AND NEGATIVE
To this point, we have been dealing with one type of LOGIC POLARITY,
positive. Let’s further define logic polarity and expand to cover in more detail the differences between positive
and negative logic.
Logic polarity is the type of voltage used to represent the logic 1 state of a
statement. We have determined that the two logic states can be represented by electrical signals. Any two distinct
voltages may be used. For instance, a positive voltage can represent the 1 state, and a negative voltage can
represent the 0 state. The opposite is also true.
Logic circuits are generally divided into two broad
classes according to their polarity — positive logic and negative logic. The voltage levels used and a statement
indicating the use of positive or negative logic will usually be specified on logic diagrams supplied by
In practice, many variations of logic polarity are used; for example, from a high-positive
to a low- positive voltage, or from positive to ground; or from a high-negative to a low-negative voltage, or from
negative to ground. A brief discussion of the two general classes of logic polarity is presented in the following
Positive logic is defined as follows: If the signal that activates the
circuit (the 1 state) has a voltage level that is more POSITIVE than the 0 state, then the logic polarity is
considered to be POSITIVE. Table 2-2 shows the manner in which positive logic may be used.
Table 2-2. — Examples of Positive Logic
As you can see, in positive logic the 1 state is at a more positive voltage level than the 0 state.
As you might suspect, negative logic is the opposite of positive logic and
is defined as follows: If the signal that activates the circuit (the 1 state) has a voltage level that is more
NEGATIVE than the 0 state, then the logic polarity is considered to be NEGATIVE. Table 2-3 shows the manner in
which negative logic may be used.
Table 2-3.—Examples of Negative Logic
NOTE: The logic level LOW now represents the 1 state. This is because the 1 state voltage is
more negative than the 0 state.
In the examples shown for negative logic, you notice that the voltage for
the logic 1 state is more negative with respect to the logic 0 state voltage. This holds true in example 1 where
both voltages are positive. In this case, it may be easier for you to think of the TRUE condition as being less
positive than the FALSE condition. Either way, the end result is negative logic.
The use of positive or
negative logic for digital equipment is a choice to be made by design engineers. The difficulty for the technician
in this area is limited to understanding the type of logic being used and keeping it in mind when troubleshooting.
UNLESS OTHERWISE NOTED, THE REMAINDER OF THIS BOOK WILL DEAL ONLY
WITH POSITIVE LOGIC.
LOGIC INPUTS AND OUTPUTS
As you study logic circuits, you will see a variety of
symbols (variables) used to represent the inputs and outputs. The purpose of these symbols is to let you know what
inputs are required for the desired output.
If the symbol A is shown as an input to a logic device, then
the logic level that represents A must be HIGH to activate the logic device. That is, it must
satisfy the input requirements of the logic device before the logic device will issue the TRUE output.
Look at view A of figure 2-1. The symbol X represents the input. As long as the switch is open, the lamp is not
lit. The open switch represents the logic 0 state of variable X.
Figure 2-1. —Logic switch: A. Logic 0 state; B. Logic 1 state.
Closing the switch (view B), represents the logic 1 state of X. Closing the switch completes the circuit
causing the lamp to light. The 1 state of X satisfied the input requirement and the circuit therefore produced the
desired output (logic HIGH); current was applied to the lamp causing it to light.
If you consider the lamp
as the output of a logic device, then the same conditions exist. The TRUE (1 state) output of the logic device is
to have the lamp lit. If the lamp is not lit, then the output of the logic device is FALSE (0 state).
you study logic circuits, it is important that you remember the state (1 or 0) of the inputs and outputs.
So far in this chapter, we have discussed the two conditions of logical statements, the logic states representing
these two conditions, logic levels and associated electrical signals and positive and negative logic. We are now
ready to proceed with individual logic device operations. These make up the majority of computer circuitry.
As each of the logic devices are presented, a chart called a TRUTH TABLE will be used to illustrate all possible
input and corresponding output combinations. Truth Tables are particularly helpful in understanding a logic device
and for showing the differences between devices.
The logic operations you will study in this chapter are
the AND, OR, NOT, NAND, and NOR. The devices that accomplish these operations are called logic gates, or more
informally, gates. These gates are the foundation for all digital equipment. They are the "decision-making"
circuits of computers and other types of digital equipment. By making decisions, we mean that certain conditions
must exist to produce the desired output.
In studying each gate, we will introduce various mathematical
SYMBOLS known as BOOLEAN ALGEBRA expressions. These expressions are nothing more than
descriptions of the input requirements necessary to activate the circuit and the resultant circuit output.
THE AND GATE
The AND gate is a logic circuit that requires all inputs to be TRUE at the same time in order
for the output to be TRUE.
The standard symbol for the AND gate is
shown in figure 2-2. Variations of this standard symbol may be encountered. These variations become necessary to
illustrate that an AND gate may have more than one input.
Figure 2-2. —AND gate.
If we apply two variables, A and B, to the inputs of the AND gate, then both A and B would have to be TRUE at
the same time to produce the desired TRUE output. . The symbol f designates the output function. The Boolean
expression for this operation is f = A·B or f = AB. The expression is spoken, "f = A AND B." The dot, or lack of,
indicates the AND function.
AND GATE OPERATION
We can demonstrate the operation
of the AND gate with a simple circuit that has two switches in series as shown in figure 2-3. You can see that
both switches would have to be closed at the same time to light the lamp (view A). Any other combination of switch
positions (view B) would result in an open circuit and the lamp would not light (logic 0).
Figure 2-3.—AND gate equivalent circuit: A. Logic 1 state; B. Logic 0 state.
Now look at figure 2-4. Signal A is applied to one input of the AND gate and signal B to the other.
T0, both inputs are LOW (logic 0) and f is LOW. At T1, A goes HIGH (logic 1); B remains LOW; and as a result, f
remains LOW. At T2, A goes LOW and B goes HIGH; f, however, is still LOW, because the proper input conditions have
not been satisfied (A and B both HIGH at the same time). At T4, both A
and B are HIGH. As a result, f is HIGH. The input requirements have been satisfied, so the output is
HIGH (logic 1).
Figure 2-4. —AND gate input and output signals.
Now let’s refer to figure 2-5. As you can see, a Truth Table and a Table
of Combinations are shown. The latter is a deviation of the Truth Table. It uses the HIGH and LOW logic levels to
depict the gate’s inputs and resultant output combinations rather than the 1 and 0 logic states. By comparing the
inputs and outputs of the two tables, you see how one can easily be converted to the other (remember, 1 = HIGH and
0 = LOW). The Table of Combinations is shown here only to familiarize you with its existence, it will not be seen
again in this book. As we mentioned earlier, the Truth Table is a chart that shows all possible combinations of
inputs and the resulting outputs. Compare the AND gate Truth Table (figure 2-5) with the input signals shown in
Figure 2-5. —AND gate logic symbol, Truth Table, and Table of Combinations.
The first combination (A = 0, B = 0) corresponds to T0 in figure 2-4; the second to T1;
the third to T2; and the last to T4. When constructing a Truth Table, you must include all
possible combinations of the inputs, including the all 0s combination.
A Truth Table representing an AND
gate with three inputs (X, Y, and Z) is shown below. Remember that the two-input AND gate has four possible
combinations, with only one of those combinations providing a HIGH output. An AND gate with three inputs has eight
possible combinations, again with only one combination providing a HIGH output. Make sure you include all possible
combinations. To check if you have all combinations, raise 2 to the power equal to the number of input variables.
This will give you the total number of possible combinations. For example:
EXAMPLE 1-AB = 22
= 4 combinations
EXAMPLE 2-XYZ = 23 = 8 combinations
As with all AND gates, all the inputs must be HIGH at the same time to produce a HIGH output. Don’t be confused
if the complement of a variable is used as an input. When a complement is indicated as an input to an AND gate, it
must also be HIGH to satisfy the input requirements of the gate. The Boolean expression for the output is
formulated based on the TRUE inputs that give a TRUE output. Here is an adage that might help you better
understand the AND gate:
In order to produce a 1 output, all the inputs must be 1. If any or all of the inputs is/are 0, then the output
will be 0.
Referring to the following examples should help you cement this concept in your mind. Remember,
the inputs, whether the original variable or the complement must be high in order for the output to be high. The
three examples given are all AND gates with two inputs. Keep in mind the Boolean expression for the output is the
result of all the inputs being HIGH.
Figure 2-5a. —AND gate logic with two inputs, Truth Table.
You will soon be able to recognize the Truth Table for the other types of logic gates without having to look at
the logic symbol.
Q1. What is defined as "the science of reasoning?"
Q2. With regard to computer logic circuits, what is meant by "complement?"
What are the complements of the following terms?
Q4. If logic 1 = -5 vdc and logic 0 = -10 vdc, what logic polarity is being used?
Q5. If logic 1 = +2 vdc and logic 0 = -2 vdc, what logic polarity is being used?
If logic 1 = -5 vdc and logic 0 = 0 vdc, what logic polarity is being used?
Q7. What is the
Boolean expression for the output of an AND gate that has R and S as inputs?
Q8. What must be
the logic state of R and S to produce the TRUE output?
Q9. How many input combinations exist
for a four-input AND gate?
THE OR GATE
The OR gate differs from the AND gate in that only ONE input has to be HIGH to produce a HIGH output. An easy
way to remember the OR gate is that any HIGH input will yield a HIGH output.
Figure 2-6shows the standard symbol for the OR gate. The number of inputs will vary according to the needs of the
Figure 2-6. —OR gate.
The OR gate may also be represented by a simple circuit as shown in figure 2-7. In the OR gate, two switches
are placed in parallel. If either or both of the switches are closed (view A), the lamp will light. The only time
the lamp will not be lit is when both switches are open (view B).
Figure 2-7. —OR gate equivalent circuit: A. Logic 1 state; B. Logic 0 state.
Let’s assume we are applying two variables, X and Y, to the inputs of an OR gate. For the circuit to produce a
HIGH output, either variable X, variable Y, or both must be HIGH. The Boolean expression for this operation is f =
X+Y and is spoken "f equals X OR Y." The plus sign indicates the OR function and should not be confused with
OR GATE OPERATION
Look at figure 2-8. At time T0, both X and
Y are LOW and f is LOW. At T1, X goes HIGH producing a HIGH output. At T2 when both
inputs go LOW, f goes LOW. When Y goes HIGH at T3, f
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