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
COMMUTATIVE LAW  the order in which terms are written does not affect their value (AB = BA, A+B = B+A). ASSOCIATIVE LAW  a simple equality statement A(BC) = ABC or A+(B+C) = A+B+C. IDEMPOTENT LAW  a term ANDed with itself or ORed with itself is equal to that term (AA = A, A+A = A). DOUBLE NEGATIVE LAW  a term that is inverted twice is equal to the term COMPLEMENTARY LAW  a term ANDed with its complement equals 0, and a term ORed with its complement equals 1 (A A = 0, A+ A = 1). LAW OF INTERSECTION  a term ANDed with 1 equals that term and a term ANDed with 0 equals 0 (A·1 = A, A·0 = 0). LAW OF UNION  a term ORed with 1 equals 1 and a term ORed with 0 equals that term (A+1 = 1, A+0 = A). DeMORGAN'S THEOREM  this theorem consists of two parts: (1) AB = A + B and (2) A + B = A · B (Look at the fourth and eighth sets of gates in table 24). DISTRIBUTIVE LAW  (1) a term (A) ANDed with an parenthetical expression (B+C) equals that term ANDed with each term within the parenthesis: A·(B+C) = AB+AC; (2) a term (A) ORed with a parenthetical expression ( B ·C) equals that term ORed with each term within the parenthesis: A+(BC) = (A+B) · (A+C). LAW OF ABSORPTION  this law is the result of the application of several other laws: A·(A+B) = A or A+(AB) = A. LAW OF COMMON IDENTITIES  the two statements A·( A + B) = AB and A+ A B = A+B are based on the complementary law. 231
Table 25.  Boolean Laws and Theorems If you wish a more detailed study of Boolean algebra, we suggest you obtain Mathematics, Volume 3, NAVEDTRA 10073A1. Q33. Boolean algebra is based on the assumption that most quantities have conditions. Q34. Boolean algebra is used primarily by to simplify circuits. SUMMARY This chapter has presented information on logic, fundamental logic gates, and Boolean laws and theorems. The information that follows summarizes the important points of this chapter. LOGIC is the development of a logical conclusion based on known information. Computers operate on the assumption that statements have two conditions  TRUE and FALSE. 232
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. NEGATIVE LOGIC 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. In DIGITAL LOGIC (positive or negative), the TRUE condition of a statement is represented by the logic 1 state and the FALSE condition is represented by the logic 0 state. LOGIC LEVELS High and LOW represent the voltage levels of the two logic states. Logic level HIGH represents the more positive voltage while logic level LOW represents the less positive (more negative) voltage. In positive logic, the HIGH level corresponds to the TRUE or 1 state and the LOW level corresponds to the FALSE or 0 state. In negative logic, the HIGH level corresponds to the FALSE or 0 state and the LOW level corresponds to the TRUE or 1 state. A BOOLEAN EXPRESSION is a statement that represents the inputs and outputs of logic gates. The AND GATE requires all inputs to be HIGH at the same time in order to produce a HIGH output. The OR GATE requires one or both inputs to be HIGH in order to produce a HIGH output. INVERTER (NOT function or negator) is a logic gate used to complement the state of the input variable; that is, a 1 becomes a 0 or a 0 becomes a 1. It may be used on any input or output of any gate to obtain the desired result. 233
The NAND GATE functions as an AND gate with an inverted output. The NOR GATE functions as an OR gate with an inverted output. When deriving the output Boolean expression of a combination of gates, solve one gate at a time. Boolean algebra is used primarily for the design and simplification of circuits. ANSWERS TO QUESTIONS Q1. THROUGH Q34. A1. Logic. A2. The opposite of the original statement. A3. a. Q, b. R, c. V, d. Z A4. Positive. A5. Positive. A6. Negative. A7. f = RS. A8. Both must be 1s (HIGH) at the same time. 234
A9. 16. A10. f = G+K+L. A11. Eight. A12. Seven. A13. XYZ A14. X + (YZ). A15. No. A16. HIGH. A17. All inputs must be HIGH. A18. A19. Low. A20. It has an inverter on the output. A21. Low. A22. R + T A23. All inputs must be low. A24. A B . A25. OR gate. A26. A27. (ABC)(DE). A28. (ABC)+(DE). A29. (R+S+T) (X+Y+Z). A30. (R+S+T)+(X+Y+Z). A31. ( JK )( M + N ). 235
A32. (AB) (M + N) ( X + Y ). A33. Two. A34. Design engineers. 236
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
