# minimal and maximal number

Let’s consider a finite non-empty set  $A\,=\,\{a_{1},\,\ldots,\,a_{n}\}$   of real numbers or an infinite but compact (i.e. bounded and closed) set $A$ of real numbers.  In both cases the set has a unique least number and a unique greatest number.

• The least number of the set is denoted by  $\min\{a_{1},\,\ldots,\,a_{n}\}$  or  $\min{A}$.

• The greatest number of the set is denoted by  $\max\{a_{1},\,\ldots,\,a_{n}\}$  or  $\max{A}$.

In both cases we have

 $\min{A}\;=\;\inf{A},$
 $\max{A}\;=\;\sup{A},$
 $\min{A}\;\leqq\;x\;\leqq\;\max{A}\quad\forall x\in A,$

where  $\inf{A}$  and  $\sup{A}$  are the infimum and supremum of the set $A$.

The $\min$ and $\max$ are set functions, i.e. they map subsets of a certain set to $\mathbb{R}$.

The $\min$ and $\max$ have the following distributive properties with respect to addition:

 $\min\{a_{1},\,\ldots,\,a_{n}\}+b\;=\;\min\{a_{1}+b,\,\ldots,\,a_{n}+b\}$
 $\max\{a_{1},\,\ldots,\,a_{n}\}+b\;=\;\max\{a_{1}+b,\,\ldots,\,a_{n}+b\}$

The minimal and maximal number of a set of two real numbers obey the formulae

 $\min\{a,\,b\}\;=\;\frac{a\!+\!b}{2}\!-\!\frac{|a\!-\!b|}{2},$
 $\max\{a,\,b\}\;=\;\frac{a\!+\!b}{2}\!+\!\frac{|a\!-\!b|}{2},$
 $\max\{a,\,b\}-\min\{a,\,b\}\;=\;|a\!-\!b|,$
 $\max\{a,\,b\}+\min\{a,\,b\}\;=\;a\!+\!b,$
 $\max\{a,\,-a\}\;=\;|a|$
 Title minimal and maximal number Canonical name MinimalAndMaximalNumber Date of creation 2014-02-15 18:33:33 Last modified on 2014-02-15 18:33:33 Owner pahio (2872) Last modified by pahio (2872) Numerical id 25 Author pahio (2872) Entry type Definition Classification msc 26B12 Classification msc 03E04 Synonym least and greatest number Related topic Infimum  Related topic Supremum Related topic UltrametricTriangleInequality Related topic GrowthOfExponentialFunction Related topic EstimatingTheoremOfContourIntegral Related topic LeastAndGreatestValueOfFunction Related topic FuzzyLogic2 Related topic ZerosAndPolesOfRationalFunction Related topic UniformConvergenceOnUnionInterval Related topic Interprime  Related topic LehmerMean Related topic Ab Defines least number Defines greatest number Defines minimal number Defines maximal number Defines set function