# Superadditive set function

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In mathematics, a **superadditive set function** is a [set function](/source/Set_function) whose value when applied to the [union](/source/Union_(set_theory)) of two [disjoint sets](/source/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](/source/Superadditivity) for real-valued functions. It is contrasted to [subadditive set function](/source/Subadditive_set_function).

## Definition

Let \Omega be a [set](/source/Set_(mathematics)) and f \colon 2^{\Omega} \rightarrow \mathbb{R} be a [set function](/source/Set_function), where 2^\Omega denotes the [power set](/source/Power_set#Representing_subsets_as_functions) of \Omega. The function *f* is *superadditive* if for any pair of disjoint subsets S,T of \Omega, we have f(S) + f(T) \leq f(S \cup T).[1]

## See also

- [Utility functions on indivisible goods](/source/Utility_functions_on_indivisible_goods)

## Citations

1. Nimrod Megiddo (1988). ["ON FINDING ADDITIVE, SUPERADDITIVE AND SUBADDITIVE SET-FUNCTIONS SUBJECT TO LINEAR INEQUALITIES"](http://theory.stanford.edu/~megiddo/pdf/Finding_supperadditiveX.pdf). Retrieved 21 December 2015.

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Adapted from the Wikipedia article [Superadditive set function](https://en.wikipedia.org/wiki/Superadditive_set_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Superadditive_set_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.