Editing Indicator function (section)

==Basic properties==
The ''indicator'' or ''characteristic'' [[function (mathematics)|function]] of a subset {{mvar|A}} of some set {{mvar|X}} [[Map (mathematics)|maps]] elements of {{mvar|X}} to the [[codomain]] <math>\{0,\, 1\}.</math>

This mapping is [[surjective]] only when {{mvar|A}} is a non-empty [[proper subset]] of {{mvar|X}}. If <math>A = X,</math> then <math>\mathbf{1}_A \equiv 1.</math> By a similar argument, if <math>A = \emptyset</math> then <math>\mathbf{1}_A \equiv 0.</math>

If <math>A</math> and <math>B</math> are two subsets of <math>X,</math> then
<math display=block>\begin{align}
\mathbf{1}_{A\cap B}(x) ~&=~ \min\bigl\{\mathbf{1}_A(x),\ \mathbf{1}_B(x)\bigr\} ~~=~ \mathbf{1}_A(x) \cdot\mathbf{1}_B(x), \\
\mathbf{1}_{A\cup B}(x) ~&=~ \max\bigl\{\mathbf{1}_A(x),\ \mathbf{1}_B(x)\bigr\} ~=~ \mathbf{1}_A(x) + \mathbf{1}_B(x) - \mathbf{1}_A(x) \cdot \mathbf{1}_B(x)\,,
\end{align}</math>

and the indicator function of the [[Complement (set theory)|complement]] of <math>A</math> i.e. <math>A^\complement</math> is:
<math display=block>\mathbf{1}_{A^\complement} = 1 - \mathbf{1}_A.</math>

More generally, suppose <math>A_1, \dotsc, A_n</math> is a collection of subsets of {{mvar|X}}. For any <math>x \in X:</math>

<math display=block> \prod_{k \in I} \left(\ 1 - \mathbf{1}_{A_k}\!\left( x \right)\ \right)</math>

is a product of {{math|0}}s and {{math|1}}s. This product has the value {{math|1}} at precisely those <math>x \in X</math> that belong to none of the sets <math>A_k</math> and is 0 otherwise. That is

<math display=block> \prod_{k \in I} ( 1 - \mathbf{1}_{A_k}) = \mathbf{1}_{X - \bigcup_{k} A_k} = 1 - \mathbf{1}_{\bigcup_{k} A_k}.</math>

Expanding the product on the left hand side,

<math display=block>\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}}. 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>\mathbb{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}}:

<math display=block>\operatorname\mathbb{E}_X\left\{\ \mathbf{1}_A(x)\ \right\}\ =\ \int_{X} \mathbf{1}_A( x )\ \operatorname{d\ \mathbb{P} }(x) = \int_{A} \operatorname{d\ \mathbb{P} }(x) = \operatorname\mathbb{P}(A).</math>

This identity is used in a simple proof of [[Markov's inequality]].

In many cases, such as [[order theory]], the inverse of the indicator function may be defined. This is commonly called the [[generalized Möbius function]], as a generalization of the inverse of the indicator function in elementary [[number theory]], the [[Möbius function]]. (See paragraph below about the use of the inverse in classical recursion theory.)