we say K
is an
-constituent
if it is a constituent for the classes
.
Boole used the following theorem to
expand
a given expression
into its constituents.
It says basically that
F is a union of -constituents, and a given -constituent
K is a constituent of this expansion of F if and
only if 
, where 
is the unique sequence 
of 1's and 0's that make 
. The following gives the
for the constituents belonging to the three classes A,B,C:
Note that if K,L are both constituents then
be given. Then
or, more briefly,
where K ranges over the 
-constituents.
Proof By using the fundamental identities on page 11 one sees that it is possible to express F as a union of constituents. So we write
where each constant 
is either 0 or 1.
We can write this more briefly as follows (using the fact that there
are 
constituents for n variables)
where the 
are the various -constituents.
Now one observes that for 
being the unique sequence of 1's and 0's
that gives 
, i.e., 
, we have
as desired.
