  # Module 13 - Introduction to Number Systems and LogicNavy Electricity and Electronics Training Series (NEETS)Chapter 2:  Pages 2-31 through 2-36

Module 13 - Introduction to Number Systems and Logic

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 2-4).

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.

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Table 2-5. - Boolean Laws and Theorems If you wish a more detailed study of Boolean algebra, we suggest you obtain Mathematics, Volume 3, NAVEDTRA 10073-A1.

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.

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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. 2-33

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.

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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 ).

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A32.   (AB) (M + N) ( X + Y ).

A33.   Two.

A34.   Design engineers.

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