# Symbols:Brackets/Round Brackets

## Symbol

Round brackets are the symbols most used for placing an expression in parenthesis.

They comprise the left (round) bracket $($ and the right (round) bracket $)$.

The $\LaTeX$ code for $($ is ( .

The $\LaTeX$ code for $)$ is ) .

## Usage

Round brackets are used in the following contexts:

To denote binding priority, for example $a \times \paren {b + c}$
To denote an ordered tuple: $\tuple {a_1, a_2, a_2, a_4}$
To contain the argument or arguments of a mapping: $\map f {x, y}$

The $\LaTeX$ code for $a \times \paren {b + c}$ is a \times \paren {b + c} .

The $\LaTeX$ code for $\tuple {a_1, a_2, a_2, a_4}$ is \tuple {a_1, a_2, a_2, a_4} .

The $\LaTeX$ code for $\map f {x, y}$ is \map f {x, y} .

### Binomial Coefficient

$\dbinom n m$

The binomial coefficient, which specifies the number of ways you can choose $m$ objects from $n$ (all objects being distinct).

Formally defined as:

$\dbinom n m = \begin {cases} \dfrac {n!} {m! \, \paren {n - m}!} & : m \le n \\ 0 & : m > n \end {cases}$

The $\LaTeX$ code for $\dbinom {n} {m}$ is \dbinom {n} {m}  or \ds {n} \choose {m}.

### Greatest Common Divisor: Deprecated Symbol

$\tuple {a, b}$

The notation for the greatest common divisor is commonly seen as:

$\tuple {a, b}$

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

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

### Open Interval

$\openint a b$

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 $b$.

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

### Open Interval: Deprecated Symbol

$\tuple {a, b}$

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

$\tuple {a, b} := \set {x \in S: a < x < b}$

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

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