# Symbols:Brackets/Square Brackets

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## Symbol

Square brackets are symbols used in a number of specific contexts.

They comprise the left (square) bracket $[$ and the right (square) bracket $]$.

The $\LaTeX$ code for $[$ is [ .

The $\LaTeX$ code for $]$ is ] .

## Usage

Square brackets are used in the following contexts:

Expressing a number to a specific base, for example: $\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$
To contain the argument for the image of a subset under a mapping: $f \sqbrk T := \set {\map f x: x \in T}$

The $\LaTeX$ code for $\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$ is \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b .

The $\LaTeX$ code for $f \sqbrk T$ is f \sqbrk T .

### Lowest Common Multiple: Deprecated Symbol

$\sqbrk {a, b}$

The notation for the lowest common multiple is commonly seen as:

$\sqbrk {a, b}$

However, as the $\sqbrk {a, b}$ notation is ambiguous, its use is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for $\sqbrk {a, b}$ is \sqbrk {a, b} .

### Closed Interval

$\closedint a b$

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 $b$.

The $\LaTeX$ code for $\closedint a b$ is \closedint a b .

### Closed Interval: Deprecated Symbol

$\sqbrk {a, b}$

The notation for a closed interval is more commonly seen as:

$\sqbrk {a, b} := \set {x \in S: a \le x \le b}$

However, as the $\sqbrk {a, b}$ notation is ambiguous, its use is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for $\sqbrk {a, b}$ is \sqbrk {a, b} .

### Right Half-Open Interval

$\hointr a b$

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 $b$.

The $\LaTeX$ code for $\hointr a b$ is \hointr a b .

### Right Half-Open Interval: Deprecated Symbol

$\left [{a, b}\right)$

The notation for a right half-open interval is more commonly seen as:

$\left [{a, b}\right) := \set {x \in S: a \le x < b}$

However, for consistency with other interval notation, its use is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for $\left [{a, b}\right)$ is \left [{a, b}\right) .

### Half-Open Interval to the Left

$\hointl a b$

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 $b$.

The $\LaTeX$ code for $\hointl a b$ is \hointl a b .

### Left Half-Open Interval: Deprecated Symbol

$\left ({a, b}\right]$

The notation for a left half-open interval is more commonly seen as:

$\left ({a, b}\right] := \set {x \in S: a < x \le b}$

However, for consistency with other interval notation, its use is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for $\left ({a, b}\right]$ is \left ({a, b}\right] .