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

$\begin{align} \max\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i<j}\min\{x_i,x_j\} +\sum_{i<j<k}\min\{x_i,x_j,x_k\} - \cdots \\ & \qquad \cdots + \left(-1\right)^{n+1}\min\{x_1,x_2,\ldots,x_n\},\end{align}$

or conversely

$\begin{align} \min\{x_1,x_2,\ldots,x_{n}\} & = \sum_{i=1}^n x_i - \sum_{i<j}\max\{x_i,x_j\} +\sum_{i<j<k}\max\{x_i,x_j,x_k\} - \cdots \\ & \qquad \cdots + \left(-1\right)^{n+1}\max\{x_1,x_2,\ldots,x_n\}. \end{align}$

For a probabilistic proof, see the reference.

See also

• Inclusion-exclusion principle
• Sorting identity

References

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