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Difference between revisions of "Interval and segment"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520902.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520904.png" /> themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520906.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520908.png" /> are included. The terms  "interval"  and "segment"  are also used for the corresponding sets of real numbers: an interval consists of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520909.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209010.png" />, while a segment consists of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209011.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209012.png" />. An interval is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209013.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209014.png" />, and a segment by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209015.png" />.
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The term  "interval"  is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209016.png" /> but also the infinite, or improper, intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209019.png" />, the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209020.png" />, and the half-open intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209024.png" />. 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.
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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  $  A $
 +
and  $  B $,
 +
where  $  A $
 +
and  $  B $
 +
themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points  $  A $
 +
and  $  B $,
 +
where  $  A $
 +
and  $  B $
 +
are included. The terms  "interval"  and  "segment"  are also used for the corresponding sets of real numbers: an interval consists of numbers  $  x $
 +
satisfying  $  a < x < b $,
 +
while a segment consists of those  $  x $
 +
for which  $  a \leq  x \leq  b $.
 +
An interval is denoted by  $  ( a , b ) $,
 +
or  $  \left ] a , b \right [ $,
 +
and a segment by  $  [ a , b ] $.
 +
 
 +
The term  "interval"  is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only $  ( a , b ) $
 +
but also the infinite, or improper, intervals $  ( - \infty , a ) $,
 +
$  ( a , + \infty ) $,  
 +
$  ( - \infty , + \infty ) $,  
 +
the segment $  [ a , b ] $,  
 +
and the half-open intervals $  [ a , b ) $,
 +
$  ( a , b ] $,
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$  ( - \infty , a ] $,  
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$  [ a , + \infty ) $.  
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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''
 
''BSE-3''
  
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]]''.
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The notion of an interval in a [[partially ordered set]] is more general. An interval $  [ a , b ] $
 +
consists in this setting of all elements $  x $
 +
of the partially ordered set that satisfy $  a \leq  x \leq  b $.  
 +
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''

Latest revision as of 22:13, 5 June 2020


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 $ A $ and $ B $, where $ A $ and $ B $ themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points $ A $ and $ B $, where $ A $ and $ B $ are included. The terms "interval" and "segment" are also used for the corresponding sets of real numbers: an interval consists of numbers $ x $ satisfying $ a < x < b $, while a segment consists of those $ x $ for which $ a \leq x \leq b $. An interval is denoted by $ ( a , b ) $, or $ \left ] a , b \right [ $, and a segment by $ [ a , b ] $.

The term "interval" is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only $ ( a , b ) $ but also the infinite, or improper, intervals $ ( - \infty , a ) $, $ ( a , + \infty ) $, $ ( - \infty , + \infty ) $, the segment $ [ a , b ] $, and the half-open intervals $ [ a , b ) $, $ ( a , b ] $, $ ( - \infty , a ] $, $ [ a , + \infty ) $. 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 $ [ a , b ] $ consists in this setting of all elements $ x $ of the partially ordered set that satisfy $ a \leq x \leq b $. 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.

How to Cite This Entry:
Interval and segment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval_and_segment&oldid=47404
This article was adapted from an original article by BSE-3, L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article