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## 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 A B. 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 C D={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. A B. 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 L E denotes the set of people with both pulmonary and extrapulmonary involvement.

On to complements.