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Interval (matematika): Béda antarrépisi

Ngarapihkeun éjahan, replaced: nyaeta → nyaéta, ea → éa (6), eo → éo using AWB
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m Ngarapihkeun éjahan, replaced: nyaeta → nyaéta, ea → éa (6), eo → éo using AWB
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Interval '''R''' kabagi kana sababaraha tipe (''a'' jeung ''b'' wilangan nyata, ''a'' < ''b''):
# (''a'',''b'') = { ''x'' | ''a'' < ''x'' < ''b'' }
# [''a'',''b''] = { ''x'' | ''a'' &le; ''x'' &le; ''b'' }
# [''a'',''b'') = { ''x'' | ''a'' &le; ''x'' < ''b'' }
# (''a'',''b''] = { ''x'' | ''a'' < ''x'' &le; ''b'' }
# (''a'',&infin;) = { ''x'' | ''x'' > ''a'' }
# [''a'',&infin;) = { ''x'' | ''x'' &ge; ''a'' }
# (-&infin;,''b'') = { ''x'' | ''x'' < ''b'' }
# (-&infin;,''b''] = { ''x'' | ''x'' &le; ''b'' }
# (-&infin;,&infin;) = '''R''' téa, sét sakabéh [[wilangan nyata]]
# {''a''}
# [[sét kosong]]
In eachéach case where they appearappéar above, ''a'' and ''b'' are known as '''endpoints''' of the interval.
Note that a square bracket [ or ] indicates that the endpoint is included in the interval, while a round bracket ( or ) indicates that it is not.
For more information about the notation used above, see [[Naive set theory]].
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Interval (10) is also known as a '''singleton'''.
The '''length''' of the bounded intervals (1), (2), (3), (4) is ''b''-''a'' in eachéach case. The '''total length''' of a [[sequence]] of intervals is the sum of the lengths of the intervals. No allowance is made for the [[Set theoretic intersection|intersection]] of the intervals. For instance, the total length of the [[sequence]] {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the [[Set theoretic union|union]] of the sequence is an interval of length 1.5.
Intervals play an important role in the theorythéory of [[integration]], because they are the simplest [[set]]s whose "size" or "measure" or "length" is easyéasy to define (see above).
The concept of measureméasure can then be extended to more complicated sets, leadingléading to the [[Borel measure]] and eventually to the [[Lebesgue measure]].
Intervals are precisely the [[connectedness|connected]] subsets of '''R'''. They are also precisely the [[convex]] subsets of '''R'''.
Since a [[continuous]] image of a connected set is connected,
it follows that if ''f'': '''R'''&rarr;'''R''' is a continuous function and ''I'' is an interval, then its image ''f''(''I'') is also an interval.
This is one formulation of the [[intermediate value theorem]].
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In [[order theory]], one usually considers [[partially ordered set]]s. However, the above notations and definitions can immediately be applied to this general case as well. Of special interest in this general setting are intervals of the form [''a'',''b''].
For a partially ordered set (''P'', &le;) and two elements ''a'' and ''b'' of ''P'', one defines the set
: [''a'', ''b''] = { ''x'' | ''a'' &le; ''x'' &le; ''b'' }
One may choose to restrict this definition to pairs of elements with the property that ''a'' &le; ''b''. Alternatively, the intervals without this condition will just coincide with the [[empty set]], which in the former case would not be considered as an interval.
== Interval arithmetic ==
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'''Interval arithmetic''', also called '''interval mathematics''', '''interval analysis''', and '''interval computations''', has been introduced in 1956 by M. Warmus. It defines a set of operations which can be applied on intervals :
T &middot;· S = { ''x'' | there is some ''y'' in ''T'', and some ''z'' in ''S'', such that ''x'' = ''y'' &middot;· ''z'' }
* [''a'',''b''] + [''c'',''d''] = [''a''+''c'', ''b''+''d'']
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==Notasi alternatip==
Cara sejen keur nuliskeun interval, ilaharna katempo di [[France]] sarta sababarha nagara Eropa sejenna, nyaetanyaéta:
* ]''a'',''b''[ = { ''x'' | ''a'' < ''x'' < ''b'' }
* [''a'',''b''] = { ''x'' | ''a'' &le; ''x'' &le; ''b'' }
* [''a'',''b''[ = { ''x'' | ''a'' &le; ''x'' < ''b'' }
* ]''a'',''b''] = { ''x'' | ''a'' < ''x'' &le; ''b'' }
==Tumbu kaluar==
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*[http://www.cs.utep.edu/interval-comp/icompwww.html Interval computations research centers]
[[CategoryKategori:Order theory]]