A set function is additive iff it is submodular and supermodular iff it is subadditive and superadditive.

Dependencies:

1. Additive set function
2. Submodular function
3. Supermodular function
4. Subadditive and superadditive set functions

Let $f: 2^{\Omega} \to \mathbb{R}$ be a set function. Then

• $f$ is additive iff it is subadditive and superadditive.
• $f$ is additive iff it is submodular and supermodular.

Proof

It's trivial to see that an additive function is submodular, supermodular, subadditive, superadditive.

Let $X$ and $Y$ be two disjoint sets. By subadditive and superadditivity (or by submodularity and supermodularity) we get $f(X \cup Y) = f(X) + f(Y)$.

Dependency for: None

Info:

• Depth: 4
• Number of transitive dependencies: 7

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function
4. Subadditive and superadditive set functions
5. Additive set function
6. Supermodular function
7. Submodular function