Indicator function

From Wikipedia, the free encyclopedia

In mathematics, an indicator function or a characteristic function is a function defined on a set $X$ that indicates membership of an element in a subset $A$ of $X$ .

The term "characteristic function" has an unrelated meaning in probability theory. For this reason, probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function to describe the function which indicates membership in a set. Another name is the representing function that Stephen Kleene (1952) (p. 227) defined in the context of the primitive recursive functions as a function φ of a predicate P that takes on values 0 if the predicate has a truth value of "true" and 1 if the predicate has a truth value of "false"; in the same context Boolos-Burgess-Jeffrey (2002) use the name "characteristic function" (defined p. 73) synonymously with Kleene's "representing function".

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

$\mathbf{1}_A : X \to \lbrace 0,1 \rbrace \,$

defined as

$\mathbf{1}_A(x) = \left\{\begin{matrix} 1 &\mbox{if}\ x \in A, \\ 0 &\mbox{if}\ x \notin A. \end{matrix}\right.$

The indicator function of $A$ is sometimes denoted

χ A (x)

or $\mathbf{I}_A(x)$ or even

A (x).

(The Greek letter χ because it is the initial letter of the Greek etymon of the word characteristic.)

The Iverson bracket allows the notation $[x \in A]$ .

Warnings.

The notation $\mathbf{1}_A$ may signify the identity function.
The notation $χ A$ may signify the characteristic function in convex analysis.

A related concept in statistics is that of a dummy variable (this must not be confused with "dummy variables" as that term is usually used in mathematics, also called a bound variable).

[edit] Basic properties

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

If $A$ and $B$ are two subsets of $X$ , then

$\mathbf{1}_{A\cap B} = \min\{\mathbf{1}_A,\mathbf{1}_B\} = \mathbf{1}_A \mathbf{1}_B,\,$

$\mathbf{1}_{A\cup B} = \max\{{\mathbf{1}_A,\mathbf{1}_B}\} = \mathbf{1}_A + \mathbf{1}_B - \mathbf{1}_A \mathbf{1}_B,$

$\mathbf{1}_{A\triangle B} = \mathbf{1}_A + \mathbf{1}_B - 2(\mathbf{1}_{A\cap B}),$

and

$\mathbf{1}_{A^\complement} = 1-\mathbf{1}_A.$

More generally, suppose $A_1, \ldots, A_n$ is a collection of subsets of $X$ . For any $x \in X$ ,

$\prod_{k \in I} ( 1 - \mathbf{1}_{A_k}(x))$

is clearly a product of $0$ s and $1$ s. This product has the value 1 at precisely those $x \in X$ which belong to none of the sets $A k$ and is $0$ otherwise. That is

$\prod_{k \in I} ( 1 - \mathbf{1}_{A_k}) = \mathbf{1}_{X - \bigcup_{k} A_k} = 1 - \mathbf{1}_{\bigcup_{k} A_k}$

Expanding the product on the left hand side,

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

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 $\mathbb{P}$ and $A$ is a measurable set, then $\mathbf{1}_A$ becomes a random variable whose expected value is equal to the probability of $A :$

$E(\mathbf{1}_A)= \int_{X} \mathbf{1}_A(x)\,dP = \int_{A} dP = P(A).\quad$

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

[edit] References

Folland, G.B.; Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 5.2: Indicator random variables, pp.94–99.
Stephen Kleene, (1952), Introduction to Metamathematics, Wolters-Noordhoff Publishing and North Holland Publishing Company, Netherlands, Sixth Reprint with corrections 1971.
George Boolos, John P. Burgess, Richard C. Jeffrey (2002), Cambridge University Press, Cambridge UK, ISBN 0521 00758 5.

Indicator function

From Wikipedia, the free encyclopedia

[edit] Basic properties

[edit] References

[edit] See also

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