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In mathematics, a superadditive set function is a set function whose value when applied to the union of two disjoint sets is greater than or equal to the sum of values of the function applied to each of the sets separately. This definition is analogous to the notion of superadditivity for real-valued functions. It is contrasted to subadditive set function.

Definition

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Let be a set and be a set function, where denotes the power set of . The function f is superadditive if for any pair of disjoint subsets of , we have .[1]

See also

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Citations

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  1. ^ Nimrod Megiddo (1988). "ON FINDING ADDITIVE, SUPERADDITIVE AND SUBADDITIVE SET-FUNCTIONS SUBJECT TO LINEAR INEQUALITIES" (PDF). Retrieved 21 December 2015.

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Set function

mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values

Superadditivity

In mathematics, a function f {\displaystyle f} is superadditive if f ( x + y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x+y)\geq f(x)+f(y)} for all x {\displaystyle

Shapley value

_{i}(v)\leq v(\{i\})} . Similarly, if v {\displaystyle v} is a superadditive set function, i.e., if v ( S ∪ T ) ≥ v ( S ) + v ( T ) {\displaystyle v(S\cup

Subadditive set function

subadditive. The maximum of additive set functions is subadditive (dually, the minimum of additive functions is superadditive). Formally, for each i ∈ { 1 ,

Supermodular function

Pseudo-Boolean function Topkis's theorem Submodular set function Superadditive Utility functions on indivisible goods Topkis, Donald M., ed. (1998). Supermodularity

Utility functions on indivisible goods

u} is a superadditive set function. Assuming u ( ∅ ) {\displaystyle u(\emptyset )} is non-positive, every supermodular function is superadditive. However

Fekete's lemma

{ a n } n = 1 ∞ {\displaystyle \{a_{n}\}_{n=1}^{\infty }} is called superadditive if and only if for all m , n ∈ N {\displaystyle m,n\in \mathbb {N} }

Sublinear function

functional – Function made from a set Norm (mathematics) – Length in a vector space Seminorm – Mathematical function Superadditivity – Property of a function Proofs