Unformatted text preview: XOR operation, and the condition whereby a true
result is obtained when both variables are true is
the distinguishing feature between the XNOR and
NOR operations. The XNOR operation having two variables can
be implemented using the logic circuit shown in
Fig. A7 (a). Other configurations are possible
depending on the logic components available, but
the logic symbol of (b) is commonly used to
indicate the XNOR function regardless of the type
of Circuit. It can be seen from the logic circuit how
the XNOR function is produced. For example, if
variables A and B both are true or false simulta
neously, the inverters insure that the inputs to both
AND gates have opposite states. Therefore, both
AND gate outputs are 0, which is inverted to 1 by the output NOR gate. If input variables A and B
have opposite states, the inverters insure that one
or the other of the AND gates has 1’s on both
inputs. This results in a 1 output from the affected
AND gate, which is inverted to 0 by the output
NOR gate. The truth table of (c) shows all possible
combinations of the XNOR gate inputoutput
conditions. The expanded version of the two variable XNOR
equation is stated as follows: X=A§+EB Mathematical manipulation of 3111.5. equation will
produce the result X = AB + AB, which can be
verified by the truth table of Fig. A7 (c). This
expression is sometimes referred to as an equality
statement and its circuit is referred to as an equality
comparator. The condensed XNOR equation form is: X=A€BB mu: XNOR A .—
B
(:53 Fig. A7 (c) Laws of Boolean Algebra Boolean logic statements rarely consist of single Appendix A A5 “ operations. They more commonly involve a number
of operations having two or more variables. This
makes them difﬁcult to visualize and the deductive
analysis process employed for basic operations
becomes cumbersome, if not impossible, as the
statements increase in complexity. In order to
implement Boolean statements mathematically,
whether simple or complex, laws have been
developed using basic expressions, the validity of
which can be proved by logical analysis. These laws
are termed the commutative, associative and
distributive laws and have identical counterparts
in ordinary mathematics. The logical AND () and
OR (+) connectives take on the connotation of
mathematical product and sum operators in these
laws, but this is done so that the principles of
algebra be applied to logic statements. Commutative Law. This law states that the
result of a product (AND) or sum (OR) operation
is unaffected by the sequence of the elements that
make up the expression. For a two element‘expres
sion using the variables A and B, two important
postulates can be stated: Postulate 13 A+B = B+A
Postulate 14 AB = BA These postulates indicate that as long as the same
logical connective is used between a given group of
elements, the positions of the elements on either
side of the equal sign may be interchanged without affecting the result. Associative Law. This law states that the
elements of a logic expression may be grouped in
any quantity provided they are connected by the
same sign. Two postulates evolve from this for the
OR and AND operations: Postulate15 A+(B+C)=(A+B)+C
Postulate 16 A(BC) = (AB)C In other words, it makes no difference how the
elements are grouped as long as the AND and OR
connectives are not changed. Note: In logic notation
involving multiple elements of the AND operation
it is common practice to omit the operational sign
() and write the terms together (e.g. ABC). Where
convenience dictates, this practice should be
followed in order to simplify the statement of AND
expressions. Distributive Law. This law states that if the
sum of two or more quantities (multiplicand) is
multiplied by another quantity (multiplier), the
product is equal to the sum of the products obtained
by multiplying each term of the multiplicand by the
multiplier. This law is purely mathematical in ...
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 Fall '05
 Myer,B

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