Definition:Interval/Ordered Set/Closed

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Definition

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

Let $a, b \in S$.


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 $b$.


Integer Interval

When $S$ is the set $\N$ of natural numbers or $\Z$ of integers, then $\closedint m n$ is called an integer interval.


The integer interval between $m$ and $n$ is denoted and defined as:

$\closedint m n = \begin{cases} \set {x \in S: m \le x \le n} & : m \le n \\ \O & : n < m \end{cases}$

where $\O$ is the empty set.


Also defined as

Some sources require that $a \preccurlyeq b$, which ensures that the interval is non-empty.


Also see

  • Results about intervals can be found here.


Technical Note

The $\LaTeX$ code for \(\closedint {a} {b}\) is \closedint {a} {b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.


Sources