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 non-empty subsets of S. Let S = {x1, x2, ..., xn}. The identity states that max { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j min { x i , x j } + ∑ i < j < k min { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 min { x 1 , x 2 , … , x n } , {\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}}} or conversely min { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j max { x i , x j } + ∑ i < j < k max { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 max { x 1 , x 2 , … , x n } .

Source: Wikipedia — Maximum-minimums identity (CC BY-SA 4.0)

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 non-empty subsets of S. Let S = {x1, x2, ..., xn}. The identity states that max { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j min { x i , x j } + ∑ i < j < k min { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 min { x 1 , x 2 , … , x n } , {\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}}} or conversely min { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j max { x i , x j } + ∑ i < j < k max { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 max { x 1 , x 2 , … , x n } .

Source: Wikipedia "Maximum-minimums identity" · CC BY-SA 4.0

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