Definition:Interval/Ordered Set/Half-Open

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Definition

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $a, b \in S$.


Left Half-Open Interval

The left half-open interval between $a$ and $b$ is the set:

$\hointl a b := a^\succ \cap b^\preccurlyeq = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }$

where:

$a^\succ$ denotes the strict upper closure of $a$
$b^\preccurlyeq$ denotes the lower closure of $b$.


Right Half-Open Interval

The right half-open interval between $a$ and $b$ is the set:

$\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$

where:

$a^\succcurlyeq$ denotes the upper closure of $a$
$b^\prec$ denotes the strict lower closure of $b$.


Wirth Interval Notation

The notation used on this site to denote an interval of an ordered set $\struct {S, \preccurlyeq}$ is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:

\(\ds \openint a b\) \(:=\) \(\ds \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }\) Open Interval
\(\ds \hointr a b\) \(:=\) \(\ds \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }\) Right Half-Open Interval
\(\ds \hointl a b\) \(:=\) \(\ds \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }\) Left Half-Open Interval
\(\ds \closedint a b\) \(:=\) \(\ds \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }\) Closed Interval


Also known as

A half-open interval can also be referred to as half-closed.


Also see

  • Results about intervals can be found here.


Sources

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