Module 13  Introduction to Number Systems and Logic
Pages i  ix,
11 to 110,
111 to 120,
121 to 133,
131 to 140,
141 to 150,
151 to 160,
161 to 69,
21 to 210, 211 to 220,
221 to 230,
231 to 236,
31 to 310,
311 to 220,
321 to 330,
331 to 340,
341 to 346, Index
Figure 222.  NOR gate with one inverted input.
Table 24 illustrates AND, NOR, NAND, and OR gate combinations that produce the same output. You can see by the
table that there is more than one way to achieve a desired output. Although the gates have only two inputs, the
table can be extended to more than two inputs.
221
Table 24.  Equivalent AND and NOR, NAND and OR Gates
Q24. What is the output Boolean expression for an AND gate with A and B as inputs when the B input
is inverted? Q25. What is the equivalent logic gate of a twoinput NAND gate with both inputs
inverted?
222
Q26. What is the output Boolean expression for the following gates?
LOGIC GATES IN COMBINATION
When you look at logic circuit diagrams for digital equipment, you are not going to see just a single gate, but
many combinations of gates. At first it may seem confusing and complex. If you interpret one gate at a time, you
can work your way through any network. In this section, we will analyze several combinations of gates and then
provide you with some practice problems. Figure 223 (view A) shows a simple combination of AND gates. The
outputs of gates 1 and 2 are the inputs to gate 3. You already know that both inputs to an AND gate must be HIGH
at the same time in order to produce a HIGH output.
223
Figure 223.  Logic gate combinations: A. Simple combination of AND gates; B. Simple combination
of AND gates and OR gate.
The output Boolean expression of gate 1 is RS, and the output expression of gate 2 is TV. These two output
expressions become the inputs to gate 3. Remember, the output Boolean expression is the result of the inputs, in
this case (RS)(TV); spoken "quantity R AND S AND quantity T AND V." In view B we have changed gate 3 to an
OR gate. The outputs of gates 1 and 2 remain the same but the output of gate 3 changes as you would expect. The
output of gate 3 is now (RS)+(TV); spoken "quantity R AND S OR quantity T AND V." In figure 224 (view A),
the outputs of two OR gates are being applied as the input to third OR gate. The output for gate 1 is R+S, and the
output for gate 2 is T+V. With these inputs, the output expression of gate 3 is (R+S)+(T+V).
224
Figure 224.  Logic gate combinations: A. Simple combination of OR gates; B. Simple combination
of OR gates and AND gate; C. Output expression without the parentheses.
In view B, gate 3 has been changed to an AND gate. The outputs of gates 1 and 2 do not change, but the output
expression of gate 3 does. In this case, the gate 3 output expression is (R+S)(T+V). This expression is spoken,
"quantity R OR S AND quantity T OR V." The parentheses are used to separate the input terms and to indicate the
AND function. Without the parentheses the output expression would read R+ST+V, which is representative of the
circuit in view C. As you can see, this is not the same circuit as the one depicted in view B. It is very
important that the Boolean expressions be written and spoken correctly. The Truth Table for the output
expression of gate 3 (view B) will help you better understand the output. When studying this Truth Table, notice
that the only time f is HIGH (logic 1) is when either or both R and S AND either or both T and V are HIGH (logic
1).
225
Now let's determine the output expression for the NOR gate in figure 225. First write the outputs of gates 1,
2, and 3:
Figure 225.  Logic gate combinations.
Since all three outputs are applied to gate 4, proceed as you would for any NOR gate. We separate each input to
gate 4 with an OR sign (+) and then place a vinculum over the entire expression. The output expression of gate 4
is:
226
When you are trying to determine the outputs of logic gates in combination, take them one gate at a time!
Now write the output expressions for the following logic gate combinations: Q27.
Q28.
Q29.
227
Q30.
Q31.
Q32.
BOOLEAN ALGEBRA
Boolean logic, or Boolean algebra as it is called today, was developed by an English mathematician, George
Boole, in the 19th century. He based his concepts on the assumption that most quantities have two possible
conditions ¾ TRUE and FALSE. This is the same theory you were introduced to at the beginning of this chapter.
Throughout our discussions of fundamental logic gates, we have mentioned Boolean expressions. A Boolean
expression is nothing more than a description of the input conditions necessary to get the desired output. These
expressions are based on Boole's laws and theorems. PURPOSE Boolean algebra is
used primarily by design engineers. Using this system, they are able to arrange logic gates to accomplish desired
tasks. Boolean algebra also enables the engineers to achieve the desired output by using the fewest number of
logic gates. Since space, weight, and cost are important factors in the design of equipment, you would usually
want to use as few parts as possible.
228
Figure 226 (view A), shows a rather complex series of gates. Through proper application of Boolean
algebra, the circuit can be simplified to the single OR gate shown in view B. Figure 227 shows the simplification
process and the Boolean laws and theorem used to accomplish it.
Figure 226.  Logic simplification: A. Complex series of gates; B. Simplified single OR gate.
229
Figure 227.  Logic circuit simplification process.
LAWS AND THEOREMS Each of the laws and theorems of Boolean algebra, along with a
simple explanation, is listed below. LAW OF IDENTITY  a term that is TRUE in one part of
an expression will be TRUE in all parts of the expression (A = A or A = A).
230
NEETS Table of Contents
 Introduction to Matter, Energy,
and Direct Current
 Introduction to Alternating Current and Transformers
 Introduction to Circuit Protection,
Control, and Measurement
 Introduction to Electrical Conductors, Wiring
Techniques, and Schematic Reading
 Introduction to Generators and Motors
 Introduction to Electronic Emission, Tubes,
and Power Supplies
 Introduction to SolidState Devices and
Power Supplies
 Introduction to Amplifiers
 Introduction to WaveGeneration and WaveShaping
Circuits
 Introduction to Wave Propagation, Transmission
Lines, and Antennas
 Microwave Principles
 Modulation Principles
 Introduction to Number Systems and Logic Circuits
 Introduction to Microelectronics
 Principles of Synchros, Servos, and Gyros
 Introduction to Test Equipment
 RadioFrequency Communications Principles
 Radar Principles
 The Technician's Handbook, Master Glossary
 Test Methods and Practices
 Introduction to Digital Computers
 Magnetic Recording
 Introduction to Fiber Optics
