minimal and maximal numberMinimalAndMaximalNumber2004-08-31 04:07:362009-10-10 09:46:59Definitionmaximal elementleast numbergreatest numberminimal numbermaximal numberset functionleastgreatest% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet'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.
\begin{itemize}
\item The least number of the set is denoted by\, $\min\{a_1,\,\ldots,\,a_n\}$\, or\, $\min{A}$.
\item The greatest number of the set is denoted by\, $\max\{a_1,\,\ldots,\,a_n\}$\, or\, $\max{A}$.
\end{itemize}
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 {\em set functions}, i.e. they map subsets of a certain set to $\mathbb{R}$.
The minimal and maximal number of a set of two real numbers obey the formulae
$$\min\{a,\,b\} \;=\; \frac{a\!+\!b}{2}\!-\!\frac{|a\!-\!b|}{2},$$
$$\max\{a,\,b\} \;=\; \frac{a\!+\!b}{2}\!+\!\frac{|a\!-\!b|}{2},$$
$$\max\{a,\,-a\} \;=\; |a|$$