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Union

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 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 or a two is {1} {2} = {1,2}. Or we may consider C={1,2,3}, and D={3,6}. Then C D={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: A A B.
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 L E denotes the set of people with tuberculosis disease of some kind, whether extrapulmonary, pulmonary, or both.

On to intersections.