Category:Examples of Intervals
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This category contains examples of Intervals in the context of Order Theory.
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 $b$.
Pages in category "Examples of Intervals"
The following 9 pages are in this category, out of 9 total.