> Home > Statistics Notes > Probability > Intersection
Intersection
If A is a set and B is a set, then we could make a
new
set from all the elements that are in A AND B
(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 and a two on a single throw
{1}
{2} = {}.
That turns out to be the empty set; it is impossible to get a
one and a two on a single toss!
Or we may consider C={1,2,3}, and D={3,6}. Then
CD={3}.
The element 3 is the only element, since it is the only element that is
in both C and D. The element 1 appears in C,
but it does not appear in D, so 1 cannot appear in the intersection,
since to be in the intersection it must be in both sets.
Notice that
A
=
.
no matter what A happens to be. Taking the intersection
of a set A
with the empty set just gives you the empty set; to be in the intersection
of the empty set with some set, an element would have to be both in the
empty set and in the other set - but nothing is in the empty set, so there
can be nothing in the intersection of the empty set with any other set.
Also notice that
if we take two sets A and B, then the intersection has
to contain elements that are in both A and B. If
something is in this intersection, then it certainly is in A;
of course there may be things in A that aren't in the
intersection, but there certainly can't be anyting in the intersection
that is not in A.
AB.
A
For instance, just above we had {3} as the intersection of {1,2,3} and
{3,6}; the set {3} is a subset of both {1,2,3} and of the set {3,6}.
Returning to the tuberculosis example,
If L denotes pulmonary disease, and
E denotes extrapulmonary disease, then
LE
denotes the set of people with both pulmonary and extrapulmonary involvement.
