ABSTRACT
Theory on logic functions where the values of variables and their functions are 0 or 1 only is called
switching theory
. Here, let us discuss the basics of switching theory. Let us denote the set of input variables by the vector expression (
x
,
x
, …,
x
). There are 2
different input vectors when each of these
n
variables assumes the value 1 or 0. An input vector (
x
,
x
, …,
x
) such that
f
(
x
,
x
, …,
x
) = 1 or 0 is called a
true (input) vector
, or a
false (input) vector
of
f
, respectively. Vectors with
n
components are often called
n
-dimensional vectors
if we want to emphasize that there are
n
components. When the value of a logic function
f
is specified for each of the 2
vectors (i.e., for every combination of the values of
x
,
x
, …,
x
),
f
is said to be
completely specified
. Otherwise,
f
is said to be
incompletely specified
; that is, the value of
f
is specified for fewer than 2
vectors. Input vectors for which the value of
f
is not specified are called
don’t-care conditions
usually denoted by “
d”
or “
*
”
, as described in Chapter 1. These input vectors are never applied to a network whose output realizes
f
, or the values of
f
for these input vectors are not important. Thus, the corresponding values of
f
need not be considered. If there exists a pair of input vectors (
x
, …,
x
, 0,
x
, …,
x
) and (
x
, …,
x
, 1,
x
, …,
x
) that differ only in a particular variable
x
, such that the values of
f
for these two vectors differ, the logic function
f
(
x
,
x
, …,
x
) is said to be
dependent on
x
. Otherwise, it is said to be
independent of
x
. In this case,
f
can be expressed without the
x
in the logic expression of
f
. If
f
is independent of
x
,
x
is called a
dummy variable
. If
f
(
x
,
x
, …,
x
) depends on all its variables, it is said to be
non-degenerate
; otherwise,
degenerate
. For example,
x
⁄
x
x
⁄
can be expressed as
x
⁄
x
without dummy variable
x
.
Given variable
