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[parent] minimal and maximal number (Definition)

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

$\displaystyle \min{A} = \inf{A},$
$\displaystyle \max{A} = \sup{A},$
$\displaystyle \min{A} \leqq x \leqq \max{A} \quad \forall x\in A,$
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 minimal and maximal number of a set of two real numbers obey the formulae

$\displaystyle \min\{a,\,b\} = \frac{a\!+\!b}{2}\!-\!\frac{\vert a\!-\!b\vert}{2},$
$\displaystyle \max\{a,\,b\} = \frac{a\!+\!b}{2}\!+\!\frac{\vert a\!-\!b\vert}{2}.$



"minimal and maximal number" is owned by pahio.
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See Also: infimum, supremum, arithmetic-geometric-harmonic means inequality, ultrametric triangle inequality, growth of exponential function, estimating theorem of contour integral, least and greatest value of function, fuzzy logic, zeros and poles of rational function, uniform convergence on union interval

Other names:  least and greatest number
Also defines:  least number, greatest number, minimal number, maximal number, set function

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Cross-references: subsets, map, supremum, infimum, closed, bounded, compact, infinite, real numbers, finite
There are 27 references to this entry.

This is version 16 of minimal and maximal number, born on 2004-08-31, modified 2007-01-13.
Object id is 6118, canonical name is MinimalAndMaximalNumber.
Accessed 8139 times total.

Classification:
AMS MSC: 03E04 (Mathematical logic and foundations :: Set theory :: Ordered sets and their cofinalities; pcf theory)
 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
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