Indicator function: Difference between revisions

Revision as of 16:10, 15 October 2021

Revision as of 11:18, 10 November 2021

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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}}{{explain|{{mvar|F}} is defined if we know what the integer n is. But what is it supposed to represent here? Are we supposed to assume that the index set I of the {{mvar|A}}{{sub|k}}s is finite and has cardinality {{mvar|n}}?|date=May 2020}}. This is one form of the principle of [[inclusion-exclusion]].

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}}: