Why does a Venn diagram with n sets divide the universe into 
regions? The number comes from the fact that each region can be
written as an intersection of n sets. Consider this Venn diagram with n=3. There are 
regions as predicted. Each region consists of those
points in some of the subsets and not in others. For example, region 3
contains points in A and in B but not in C. That's the set 
. Region 8 is 
since it
contains points not in A, not in B and in C. So any triple
intersection involving A or 
, B or 
and C or
corresponds to one of the regions. How many such triple
intersections are there? We have 2 choices for the first set (A or ), 2 choices for the second and 2 for the third. That makes a
total of 
triple intersections. Here is the list of all
8.
There is nothing special about using 3 sets. The same reasoning applies to n
sets. If we have n sets 
and form intersections of n sets
by picking 
or 
, 
or 
and so on up
to 
or 
we will such intersections.