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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 2<sup>''n''</sup> − 1 nonempty subsets of ''S''. |
| | |
| | Let ''S'' = {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>}. The [[identity (mathematics)|identity]] states that |
| | |
| | :<math>\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}</math> |
| | or conversely |
| | |
| | :<math>\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} |
| | </math> |
| | |
| | <!-- |
| | OK, this didn't go as planned |
| | ==Inductive proof== |
| | Clearly the identity holds for ''n'' = 1. Assume it holds for ''n'' and let |
| | :<math> |
| | M_n = \max(x_1,x_2,\ldots,x_n) |
| | </math> |
| | Then for arbitrary ''x''<sub>''n''+1</sub>, |
| | :<math> |
| | M_{n+1} = \max(x_{n+1}, M_n) |
| | </math> |
| | and |
| | :<math> |
| | M_{n+1} = x_{n+1} + M_n - \min(x_{n+1}, M_n) |
| | </math> |
| | By the inductive hypothesis, |
| | :<math> |
| | M_{n+1} = x_{n+1} + \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+\left(-1\right)^{n+1}\min(x_1,x_2,\ldots,x_n) - \min(x_{n+1}, \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+\left(-1\right)^{n}\min(x_1,x_2,\ldots,x_n)) |
| | </math> |
| | collecting terms and applying the fact that the minimum operator is [[distributive property|distributive]], |
| | :<math> |
| | M_{n+1} = \sum_{i=1}^{n+1} x_i - \sum_{i<j}\min(x_i,x_j) +\sum_{i<j<k}\min(x_i,x_j,x_k) +\cdots+\left(-1\right)^{n+1}\min(x_1,x_2,\ldots,x_n) - \min(x_{n+1}, \sum_{i=1}^n x_i) + \min(x_{n+1}, \sum_{i<j}\min(x_i,x_j)) - \min(x_{n+1},\sum_{i<j<k}\min(x_i,x_j,x_k) )+\cdots+\min(x_{n+1}, \left(-1\right)^{n+1}\min(x_1,x_2,\ldots,x_n)) |
| | </math> |
| | --> |
| | For a probabilistic proof, see the reference. |
| | |
| | == See also == |
| | |
| | * [[Inclusion-exclusion principle]] |
| | * [[Sorting identity]] |
| | |
| | ==References== |
| | * {{cite book | last = Ross | first = Sheldon | title = A First Course in Probability | publisher = Prentice Hall | location = Englewood Cliffs | year = 2002 | isbn = 0-13-033851-6 }} |
| | |
| | [[Category:Mathematical identities]] |
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
or conversely
For a probabilistic proof, see the reference.
See also
References
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