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Characteristic (indicator) function

Given a set typeset structure, its characteristic function is defined by

1 _ A : A -> {0, 1},    1 _ A (x) = {1, if x ∈ A, 0, if x ∉ A .(1)

Characteristic function of the set  typeset structure:

[Graphics:HTMLFiles/IndicatorFunction_4.gif]

Since the term characteristic function has an unrelated meaning in some other branches of mathematics (for instance, in probability theory), the term indicator function for the function defined above is also used to avoid ambiguity if necessary.

The indicator function of typeset structure is also denoted by typeset structure or typeset structure. The Greek letter typeset structure goes back to the Greek typeset structure.

Using the Iverson’s bracket it can be also written in the form typeset structure.

The cardinality of a finite set typeset structure is the sum of the indicator function values  typeset structure.

If typeset structure then typeset structure for each typeset structure.

If typeset structure are two subsets of a set typeset structure, then

FormBox[RowBox[{1 _ (A ∩ B) = min {1 _ A, 1 _ B} = 1 _ A · 1 _ B, , ,, Cell[]}], TraditionalForm]

1 _ (A ∪ B) = max {1 _ A, 1 _ B} = 1 _ A + 1 _ B - 1 _ A · 1 _ B,

1 _ AΔB = 1 _ A + 1 _ B - 2 · 1 _ A · 1 _ B,

1 _ (X \ A) = 1 _ X - 1 _ A .

1 _ (A × B) (x, y) = 1 _ A (x) · 1 _ B (x)

If typeset structure is a collection of subsets of typeset structure, then we have

Underoverscript[∏, k = 1, arg3] (1 _ X - 1 _ A _ k) = 1 _ (X - Underscript[∪, k] A _ k) = 1 _ X - 1 _ (Underscript[∪, k] A _ k) .

Expanding the left hand side this yields one form of the inclusion-exclusion principle

1 _ (Underscript[∪, k] A _ k) = 1 - Underoverscript[∑, F ⊆ {1, 2, ..., n}, a ... = F ⊆ {1, 2, ..., n}, arg3] (-1)^(| F | + 1) 1 _ (Underscript[∩, k ∈ F] A _ k) ,

where typeset structure denotes the cardinality of typeset structure.

In the theory of multisets the value of the indicator function at an element of the underlying set gives its multiplicity.

In fuzzy set theory, indicator or characteristic functions are allowed to take values in the real unit interval typeset structure, or in some more general algebraic structures (e.g. a poset or a lattice).

Cite this web-page as:

Štefan Porubský: Characteristic (indicator) function.

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