Sets
 ======

  Definition and Operations
 ---------------------------

The mathematical concept of a set is an unordered collection
of unique values.  We can perform several operations with
sets including determining membership in a set (is the given
value an element inside the collection?).  We can also take
the intersection of two sets which is the collection of
elements that they have in common.  And we can take the
union of two sets which is a new collection that contains
the unique elements from both sets together.  Finally, we
can take the set difference or elements in one set but not
in the other.

  Implementation
 ----------------

To implement this all efficiently, we take the collection
and sort it from the outset.  This seems counterintuitive
since the set is supposed to be unordered by definition.
But, consider that if the set is unordered, any order will
do for representation.  And even mathameticians often list
the elements of a set manually in some sort of order to make
it easier to keep track of things.

With the data sorted, we can implement things like set
membership in logarithmic time instead of linear time, for
instance.  The other operations are sped up as well.

Ordered representations of sets are often coded as search
trees but these are not the only way.  Others use hash
tables to store sets in an unordered way.

  Disjoint Sets
 ---------------

A related concept is that of set disjointedness.  Two sets
are considered disjoint if their intersection is the empty
set.  That is, they have no overlapping elements.  The
disjoint set data structure can be used in its own right for
many interesting things and will be studied separately.