Definition:Interval/Ordered Set

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Definition

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

Let $a, b \in S$.


The intervals between $a$ and $b$ are defined as follows:


Open Interval

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

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

where:

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


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 $b$.


Closed Interval

The closed interval between $a$ and $b$ is the set:

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

where:

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


Endpoint

The elements $a, b \in S$ are known as the endpoints (or end points) of the interval.

$a$ is sometimes called the left hand endpoint and $b$ the right hand end point of the interval.


Wirth Interval Notation

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

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


Also see

  • Results about intervals can be found here.


Sources