Unformatted text preview: A2 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. A1 (a). Double complementation
converts a logic statement back to its original form
as shown in Fig. A1 (b). Postulate 3 A a9 A
Postulate 4 A = A
A—>>—n A—DD—K—cD—i=A
lol It:
Fig. Al 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 AA = A (not A2) Postulate 6 AA = 0
Postulate 7 A1 = A
Postulate 8 AO = 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 viceversa). 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. A2 (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: ABC = X A
INPUTS{ X=AB
(a) Fig. A2 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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 Fall '05
 Myer,B

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