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Relates the maximum element of a set of numbers and the minima of its non-empty subsets
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 non-empty subsets of S .
Let S = {x 1 , x 2 , ..., x n identity states that
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{\displaystyle {\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}}}
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{\displaystyle {\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.
Ross, Sheldon M. (2020). A First Course in Probability (Tenth, global ed.). Harlow, United Kingdom: Pearson. pp. 331– 333. ISBN 978-1-292-26920-7
