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→Basic properties: Repair last edit: Remove fragment of "Explain" tag. I think the sentence "More generally, suppose A_1, ..., A_n is a collection of subsets of X." makes it clear that the collection of A_k's is indeed meant to be finite, with n the number of elements of this collection. |
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: <math> \mathbf{1}_{\bigcup_{k} A_k}= 1 - \sum_{F \subseteq \{1, 2, \dotsc, n\}} (-1)^{|F|} \mathbf{1}_{\bigcap_F A_k} = \sum_{\emptyset \neq F \subseteq \{1, 2, \dotsc, n\}} (-1)^{|F|+1} \mathbf{1}_{\bigcap_F A_k} </math>
where <math>|F|</math> is the [[cardinality]] of {{mvar|F
As suggested by the previous example, the indicator function is a useful notational device in [[combinatorics]]. The notation is used in other places as well, for instance in [[probability theory]]: if {{mvar|X}} is a [[probability space]] with probability measure <math>\operatorname{P}</math> and {{mvar|A}} is a [[Measure (mathematics)|measurable set]], then <math>\mathbf{1}_A</math> becomes a [[random variable]] whose [[expected value]] is equal to the probability of {{mvar|A}}:
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