Maximum-minimums Identity
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the maximum-minimums identity is a relation between the maximum element of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'' of ''n'' numbers and the minima of the 2''n'' − 1
non-empty In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by inclu ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of ''S''. Let ''S'' = . The
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
states that :\begin \max\ & = \sum_^n x_i - \sum_\min\ +\sum_\min\ - \cdots \\ & \qquad \cdots + \left(-1\right)^\min\,\end or conversely :\begin \min\ & = \sum_^n x_i - \sum_\max\ +\sum_\max\ - \cdots \\ & \qquad \cdots + \left(-1\right)^\max\. \end For a probabilistic proof, see the reference.


See also

*
Inclusion–exclusion principle In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \cup ...
* Maxima and minima#In relation to sets


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 , url-access = registration , url = https://archive.org/details/firstcourseinpro00ross Mathematical identities