A Symbolic Analysis of Relay and Switching Circuits

41
B'! proper selection of the varialbes many
apparently uns~rmmetric functions may te made symmetric.
For example, XY'Z + X'YZ 1-.
X'Y'Z', although not symmetric
in X~
Y, end Z, is symnetl"io in X, Y, and Z'.
~e set of ~mbers a , a , ••••sk will for convenience
be called the 8-n'umbers of the function.
l 2
The theorems concerning comtlnations of symmetric
functions ere most easily stated in terms of the
018 s sa S 0 f 8 -numbar s • For thi s rea son we dena ta the
cless of a-numbers by a s1n~le letter A. If two different
sets of a-numbers are under consideration they will
be denoted by A and A 1 2
• The symmetric function of n
varia ble s heving the a -numbel'S 8 1 ,
82 •••sk will 'be
written Sn{a l
, 8 2 •••a k
) or an(A).
Theorem: 3 n (A l )· Sn(A 2 ) = Sn(A l
+ A 2 )
where A + A 1 2 means the l06!ca l sum or the classes Al
and A 2
i.e., the cla ss of tho sa numbers which B re members
of either A or A l 2 or both. Thus 36(1, 2, 3). 8 6 (2, 3, 5)
is equa1 to S6(1, 2, 3, 5).
Theorem: 3 n (A l ) + 5n(A a) ~ Sn(A 1 ,A2)
where AlwA 2 is the logical product of the mlassas i.e.,
the ala ss 0 f numbers Which are common to A 1
and A2. Thus
5 (1, 2, 3) 6
+ S6 (2, 3, 5) '"C 36(2, 3).
These theorems follow from the fact that 8
product is