1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sum of superadditive functions

  1. Feb 16, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that the sum of two superadditive (supermodular) functions is superadditive.

    2. Relevant 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)

    3. 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)]
    f(x1,y1) + f(x2,y2) >= f(x1,y2) + f(x2,y1)

    Thus, the sum of two supermodular functions is supermodular.
  2. jcsd
  3. Feb 16, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    It's better to write it out with the f's:
    [tex]f(x_1,y_1)+f(x_2,y_2)=\left(g(x_1,y_1)+h(x_1,y_1)\right)+\left(g(x_2,y_2)+h(x_2,y_2)\right) ... \geq f(x_1,y_2)+f(x_2,y_1)[/tex]
  4. Feb 16, 2007 #3
    Thank you

    Thank you so much.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook