Indicator function

From Academic Kids

In the mathematical subfield of set theory, the indicator function is a function defined on a set X which is used to indicate membership of an element in a subset A of X.

Remark. The indicator function is sometimes also called characteristic function, although this usage is much less frequent now. The term "characteristic function" is also used in probability theory where it has an entirely different meaning; see characteristic function.

The indicator function of a subset A of a set X is a function

<math>1_A : X \to \lbrace 0,1 \rbrace<math>

defined as

<math>1_A(x) =

\left\{\begin{matrix} 1 &\mbox{if}\ x \in A \\ 0 &\mbox{if}\ x \notin A \end{matrix}\right. <math>

The indicator function of A is sometimes denoted

<math>\chi_A(x) \qquad \mbox{or} \qquad I_A(x).\,<math>

Basic properties

The mapping which associates a subset A of X to its indicator function 1A is injective; its range is the set of functions f:X →{0,1}.

If A and B are two subsets of X, then

<math>1_{A\cap B} = \min\{1_A,1_B\} = 1_A 1_B \qquad \mbox{and} \qquad 1_{A\cup B} = \max\{{1_A,1_B}\} = 1_A + 1_B - 1_A 1_B.<math>

More generally, suppose A1, ..., An is a collection of subsets of X. For any xX,

<math> \prod_{k \in I} ( 1 - 1_{A_k}(x))<math>

is clearly a product of 0s and 1s. This product has the value 1 at precisely those xX which belong to none of the sets Ak and is 0 otherwise. That is

<math> \prod_{k \in I} ( 1 - 1_{A_k}) = 1_{X - \bigcup_{k} A_k} = 1 - 1_{\bigcup_{k} A_k}<math>

Expanding the product on the left hand side,

<math> 1_{\bigcup_{k} A_k}= 1 - \sum_{F \subseteq \{1, 2, \ldots, n\}} (-1)^{|F|} 1_{\bigcap_F A_k} = \sum_{\emptyset \neq F \subseteq \{1, 2, \ldots, n\}} (-1)^{|F|+1} 1_{\bigcap_F A_k} <math>

where |F| is the cardinality of 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 X is a probability space with probability measure P and A is a measurable set, then 1A becomes a random variable whose expected value is equal to the probability of A:

<math>E(1_A)= \int_{X} 1_A(x)\,dP = \int_{A} dP = P(A).\quad <math>

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

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