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Electronic Labs_9 - A-2 Appendix A postulates form the...

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Unformatted text preview: A-2 Appendix A # postulates form the fundamental definition of a two valued variable, which exists throughout Boolean algebra. The NOT Function. The NOT function, as stated previously, is a complementation or negation process. This operation causes the logic statement to assume the alternate value or inverse condition of the original statement. NOT notation is most commonly indicated by a bar symbol (-) placed over the statement. For example, if the NOT operation is applied to variable A, then A NOT is written A. If variable A is assigned the value of 1, it follows that A is equal to 0, for if A is not 1, the only other value it can have is 0. The NOT operation can be implemented by any electrical circuit that can perform an inversion process such as the inverter shown in Fig. A-1 (a). Double complementation converts a logic statement back to its original form as shown in Fig. A-1 (b). Postulate 3 A a9 A Postulate 4 A = A A—|>>—n A—DD—K—cD—i=A lol It: Fig. A-l The AND Function. When a logic statement de— pends on the simultaneous occurrence of two or more conditions, the logical AND function is necessary. The word "AND" is used in a logical sense and is not intended to imply a mathematical operation. However, the AND operation is com- monly indicated by the multiplication sign (-) used in mathematics. All logical variations of the AND function can be stated in terms of two variables using the four possible combinations of the 1 and 0 elements. These are: 1AND1=1 1ANDO=O 0AND1=0 OANDO=0 Obviously, for an AND statement to yield a true result, all of the elements must be true. If any or all elements are false, the statement is completely false. This allows four identities to be stated for the single variable occurring in coincidence with itself, its complement, a 1 and a 0. These identities are basic in appearance but are powerful tools in the manipulation of logic terms in complex equations. Postulate 5 A-A = A (not A2) Postulate 6 AA = 0 Postulate 7 A1 = A Postulate 8 A-O = 0 Postulate 5 indicates that regardless of whether the element A is a 1 or a 0, the result will be identical since the two elements are identical. Postulate 6 says that when an element and its complement are combined the result must be 0 since both cannot have the same value simulta- neously (if A is 1, A is 0 and vice-versa). Postulate 7 states that when element A and a true statement (1) coincide, the result will always be determined by the state of A. For example, if A is true the result is 1; if A false the result is O. Postulate 8 indicates that regardless of whether A is a 1 or a O, the result will always be false (0) since the one element is O. The AND operation is implemented using the logical AND gate shown in Fig. A-2 (a). It can be seen from the truth table of (b) how the funda- mental true and false relationships as well as the results of Postulates 5 through 8 are obtained. The AND operation applies to any number of variables and is stated in general form as follows: A-B-C- = X A INPUTS{ X=A-B (a) Fig. A-2 The OR Function. When a logic statement de- pends upon or consists of the independent or combined occurrence of a number of elements, the logical OR function is indicated. The OR operation is usually signified by the addition symbol (+), although it should be understood that this sign does not imply mathematical addition. Using all possible combinations of the variables 1 and 0, four logical OR statements can be made: 10R1=1 10R0=1 OOR1=1 OOR0=O It is evident that the OR function yields a true statement when one or more elements are true; the result is false only when all of the elements are false. Four very useful identities can be stated for the OR function using the variable A in relation to itself, its complement, a 1 and a 0. ...
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