The notions of union and intersection can be easily extended to more than two sets.
For finitely many sets, say 
, we write 
or 
and 
or 
. If we have a
sequence of sets 
we write 
and 
Perhaps the most general form of
this notation is a collection of sets with subscripts in some general index
set 
, 
. Here could be
finite or infinite. We then write 
and 
. So an
alternate way to write 
would be 
where N is the set of natural numbers.
Exactly
what these symbols mean is as follows.
By we mean 
and
by we mean 
Check that these become our original
definitions of set union and intersection when the indexing set has just two
elements, and then we can just call the two sets A and B. An example of
a large indexing set would be the set of reals R. If 
then 
would be the entire Cartesian
plane 
written as a union of all vertical lines in the plane.
Some care must be taken when dealing with negations involving union and
intersections. Since 
means there is at least one 
so that 
, then 
means x is not in
any 
. (Logically we have 
)
Similarly, Since
means for every , then means there is at least one with 
. (Logically we have 
)