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^''n'' − 1

non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...

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), ''Identity'' (1987 film), an Iranian film * Identity ...

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.

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

See also

References