- #1

- 73

- 0

## Homework Statement

Show that the sum of two superadditive (supermodular) functions is superadditive.

## Homework Equations

Let X and Y be partially ordered sets and g(x,y) a real-valued function on XxY. g is supermodular (superadditive) if for x1>=x2 in X and y1>=y2 in Y,

g(x1,y1) + g(x2,y2) >= g(x1,y2) + g(x2,y1)

## The Attempt at a Solution

Let g(x,y) and h(x,y) be supermodular functions on XxY. Then the following inequalities hold:

g(x1,y1) + g(x2,y2) >= g(x1,y2) + g(x2,y1)

h(x1,y1) + h(x2,y2) >= h(x1,y2) + h(x2,y1)

Let f(x,y) = g(x,y) + h(x,y), then

[g(x1,y1) + h(x1,y1)] + [g(x2,y2) + h(x2,y2)] >= [g(x1,y2) + h(x1,y2)] + [g(x2,y1) + h(x2,y1)]

implies:

f(x1,y1) + f(x2,y2) >= f(x1,y2) + f(x2,y1)

Thus, the sum of two supermodular functions is supermodular.