If A is a set and B is a set, then we can make a new
set from all the elements that are in A or B or both
(and nothing else).
This new set is written
AB.
For example, suppose we were going to consider the experiment of tossing
a single die. Then the sample space is {1,2,3,4,5,6}. The event of
getting one spot is {1}, and the event of getting two is {2}. The event
of getting a one or a two is
{1}
{2} = {1,2}.
Or we may consider C={1,2,3}, and D={3,6}. Then
CD={1,2,3,6}.
The element 3 appears only once; it must be in the union since it appears
in at least one of the sets C, D. But there is no definition
of an element appearing more than once in a set; all you can say is that
an element is in a set or it isn't.
Notice that
A = A
no matter what A happens to be. Taking the union of a set A
with the empty set just gives you the same set A. Also notice that
if we take two sets A and B, then since the union of A
and B has all the elements in A as well as those in B, then everything in
A has to be in the union of A and B. That union
may have some other things in it, but it certainly has everything in A:
AAB.
As another example, we may consider sampling a patient suspected of
having tuberculosis disease. A person may have pulmonary disease, or
extrapulmonary disease. If L denotes pulmonary disease, and
E denotes extrapulmonary disease, then
LE
denotes the set of people with tuberculosis disease of some kind, whether
extrapulmonary, pulmonary, or both.
