Set-Input Function

A Set-Input Function is a function whose function domain is from a Set Space.

• Context:
  • It can range from being, based on the input size, a Unary Set Function, Binary Set Function, an N-ary Set Function.
  • It can range from being, based on the output type, a Continuous-Output Set Function to being an Ordinal-Output Numeric Function to being a Categorical-Output Set Function.
  • It can be instantiated as a Set Function Structure.
• Example(s):
  • a Unary Set Function, such as the Set Cardinality Function.
  • the Event Probability Function.
  • a Set Relation, such as the Subset Relation.
  • a Transformation Function.
  • a Modular Function, Submodular Function, Supermodular Function.
  • …
• Counter-Example(s):
  • a Vector-Input Function.
  • a Number-Input Function.
  • a Set-Output Function.
  • a Set Operation.
  • a Multiset Function.
• See: Set, Measure Space, Normalized Set Function.

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References

2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/set_function Retrieved:2013-12-13.
  • In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input is a set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.

• http://en.wikipedia.org/wiki/Set_function#Examples
  • Examples of set functions include:
    • The function that assigns to each set its cardinality, i.e. the number of members of the set, is a set function.
    • The function: [math]\displaystyle{ d(A) = \lim_{n\to\infty} \frac{|A \cap \{1,\dots,n\}|}{n}, }[/math] assigning densities to sufficiently well-behaved subsets A ⊆ {1, 2, 3, ...}, is a set function.
    • The Lebesgue measure is a set function that assigns a non-negative real number to each set of real numbers. (Kolmogorov and Fomin 1975)
    • A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1.
    • A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See Possibility theory.

2006

• Suzanne R. Dubnicka. (2006). “STAT 510: Handout 1 - Probability Terminology. Kansas State University
  • Given a nonempty sample space S, the measure P(A) is a set function satisfying three properties.

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