Difference between revisions of "Interval and segment"
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− | The notion of an interval in a partially ordered set is more general. An interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209025.png" /> consists in this setting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209026.png" /> of the | + | The notion of an interval in a [[partially ordered set]] is more general. An interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209025.png" /> consists in this setting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209026.png" /> of the partially ordered set that satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209027.png" />. An interval in a partially ordered set that consists of precisely two elements is called a ''simple'' or an ''[[elementary interval]]''. |
''L.A. Skornyakov'' | ''L.A. Skornyakov'' | ||
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====Comments==== | ====Comments==== | ||
In English the term "segment" is not often used, except in specifically geometrical contexts; the normal terms are "open interval" and "closed interval" , cf. also [[Interval, open|Interval, open]]; [[Interval, closed|Interval, closed]]. | In English the term "segment" is not often used, except in specifically geometrical contexts; the normal terms are "open interval" and "closed interval" , cf. also [[Interval, open|Interval, open]]; [[Interval, closed|Interval, closed]]. | ||
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Revision as of 22:02, 12 October 2014
The simplest sets of points on the line. An interval (open interval) is a set of points on a line lying between two fixed points and
, where
and
themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points
and
, where
and
are included. The terms "interval" and "segment" are also used for the corresponding sets of real numbers: an interval consists of numbers
satisfying
, while a segment consists of those
for which
. An interval is denoted by
, or
, and a segment by
.
The term "interval" is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only but also the infinite, or improper, intervals
,
,
, the segment
, and the half-open intervals
,
,
,
. Here, a round bracket indicates that the appropriate end of the interval is not included, while a square bracket indicates that the end is included.
BSE-3
The notion of an interval in a partially ordered set is more general. An interval consists in this setting of all elements
of the partially ordered set that satisfy
. An interval in a partially ordered set that consists of precisely two elements is called a simple or an elementary interval.
L.A. Skornyakov
Comments
In English the term "segment" is not often used, except in specifically geometrical contexts; the normal terms are "open interval" and "closed interval" , cf. also Interval, open; Interval, closed.
Interval and segment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval_and_segment&oldid=33603