Union

The union of two sets and is the set obtained by combining the members of each. This is written , and is pronounced " union " or " cup ." The union of sets through is written . The union of a list may be computed in the Wolfram Language as Union[l].

Let , , , ... be events in a common probability space, and let denote the probability of . Then

(1)

Similarly,

			(2)
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The general statement of this property for events is known as the inclusion-exclusion principle.

If and are disjoint sets, then by definition , so

(7)

More generally, for pairwise disjoint events , , ..., , finite additivity gives

(8)

The countable additivity probability axiom extends this to a sequence of pairwise disjoint events,

(9)

If , , ..., are pairwise disjoint events and is any event, then the intersections are also pairwise disjoint. Applying finite additivity to them gives

(10)

The same identity holds for a sequence of pairwise disjoint events when both upper limits are replaced by (Hunter 2011, p. 5).