Maximum-minimums identity

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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, ..., xn}. The identity states that

\begin{align} \max\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i

or conversely

\begin{align} \min\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i

For a probabilistic proof, see the reference.

References

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