Maximum-minimums identity

In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2ⁿ − 1 nonempty subsets of S.

Let S = {x₁, x₂, ..., x_n}. The identity states that

${\begin{aligned}\max\{x_{1},x_{2},\ldots ,x_{{n}}\}&=\sum _{{i=1}}^{n}x_{i}-\sum _{{i<j}}\min\{x_{i},x_{j}\}+\sum _{{i<j<k}}\min\{x_{i},x_{j},x_{k}\}-\cdots \\&\qquad \cdots +\left(-1\right)^{{n+1}}\min\{x_{1},x_{2},\ldots ,x_{n}\},\end{aligned}}$

or conversely

${\begin{aligned}\min\{x_{1},x_{2},\ldots ,x_{{n}}\}&=\sum _{{i=1}}^{n}x_{i}-\sum _{{i<j}}\max\{x_{i},x_{j}\}+\sum _{{i<j<k}}\max\{x_{i},x_{j},x_{k}\}-\cdots \\&\qquad \cdots +\left(-1\right)^{{n+1}}\max\{x_{1},x_{2},\ldots ,x_{n}\}.\end{aligned}}$

For a probabilistic proof, see the reference.

See also

• Inclusion-exclusion principle
• Zaskulnikov's identity

References

• Ross, Sheldon (2002). A First Course in Probability. Englewood Cliffs: Prentice Hall. ISBN 0-13-033851-6.