Parentheses, Brackets and Braces

Parentheses (), brackets [], and braces {} are fundamental grouping symbols in mathematics. They clarify the order of operations, structure expressions, and organize sets, functions, and matrices. This section explains their specific purposes and how they ensure mathematical expressions are precise and unambiguous.

Parentheses

Parentheses are one of the most commonly used grouping symbols in mathematics. They are primarily used to control the order of operations, group terms, and enhance clarity.

Standard Parentheses

Enter () to group terms and clarify the order of operations.

Scaled Parentheses

Standard parentheses may not scale automatically for larger expressions, leading to visual and structural inconsistencies. Using \left( and \right) ensures that parentheses are dynamically sized to match the content they enclose.

\documentclass{article}

\begin{document}

% Without scaling: standard parentheses using ()
Standard Parentheses (no scaling):
\[
( \frac{x^2}{y^2 + z^2} )
\]

% With scaling: using \left and \right commands
Scaled Parentheses:
\[
\left( \frac{x^2}{y^2 + z^2} \right)
\]

\end{document}

This example generates the following output:

Nested Parentheses

Nested parentheses occur when one set of parentheses is placed inside another to group expressions hierarchically, ensuring clarity and controlling the order of operations in complex mathematical formulas.

The following example demonstrates how parentheses can improve clarity by explicitly showing the intended grouping of operations, even when they do not change the result:

\documentclass{article}
\begin{document}

% Same result as x+y⋅z+w, but clearer due to explicit grouping of y⋅z.
\[
(x + (y \cdot z) + w) 
\]

\end{document}

This example generates the following output:

Here’s another example where nested parentheses are required because they directly affect the result of the expression:

\documentclass{article}
\begin{document}

\[
\left( \left( x + y \right) \cdot z \right) + w 
\]

\end{document}

This example generates the following output:

Brackets

Brackets, denoted as square brackets [], are used in mathematics to organize and group terms or operations, similar to parentheses, but often to distinguish levels of nesting or to convey specific meanings in different contexts.

LaTeX syntax for brackets

• Standard brackets → Use [] directly.
• Scaled brackets → Use \left[ and \right] commands for proper sizing.

Common uses of brackets

1. Nested grouping:
Brackets are often used in combination with parentheses to clarify multiple levels of grouping:

\[
\left[ (x + y) \cdot z \right] + w 
\]

Output:

2. Matrix representation:
Brackets are commonly used to enclose elements of a matrix to define its structure.

\[
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
\]

Output:

3. Interval notation:
Brackets indicate closed intervals, where the endpoints are included.

\[
[a, b] \quad \text{(includes } a \text{ and } b\text{)}
\]

Output:

4. Combinatorial expressions:
Brackets can denote specific mathematical operations, such as Iverson brackets or floor and ceiling functions.

\[
\left[ P(x) \right] \quad \text{(Iverson Bracket: 1 if true, 0 if false)}
\]

Output:

Braces

Braces {}, also known as curly braces, are used in mathematics for various purposes, primarily to denote sets and group multiple elements or conditions, also for defining collections or functions.

LaTeX syntax for braces

• Standard braces: In LaTeX, braces have a special meaning and must be escaped when used as symbols. Simply use \{ for the left brace and \} for the right brace.

• Scaled braces: To scale the braces to fit the content inside, use \left\{ and \right\}.

Common uses of braces:

1. Set notation:
Braces are used to define a set of elements:

\[
A = \{1, 2, 3, 4\}
\]

Output:

2. Set builder notation:
Braces are used to describe a set by specifying a property that its members must satisfy:

\[
B = \{ x \in \mathbb{R} \mid x > 0 \}
\]

Output:

3. Piecewise functions:
Braces are used to group the different cases of a piecewise function:

\[
f(x) = 
\begin{cases} 
x^2 & \text{if } x > 0 \\
0 & \text{if } x \leq 0
\end{cases}
\]

Output:

Other commonly used math grouping symbols

This table highlights additional math grouping symbols commonly used in LaTeX:

Grouping Symbol	Code Example	Rendered Output
Angle brackets	\[ \langle a + b \rangle \]
Vertical bars (Absolute value)	\[ | a + b | \]
Double vertical bars (Norm)	\[ | a + b | \]
Floor function	\[ \lfloor a + b \rfloor \]
Ceiling function	\[ \lceil a + b \rceil \]

Scaling commands for delimiters in LaTeX

The commands \big, \Big, \bigg, and \Bigg in LaTeX are used to scale delimiters like parentheses, brackets, braces, and other grouping symbols. These commands ensure that the size of the delimiters matches the size of the enclosed content, improving readability in complex mathematical expressions.

Delimiter Type	Small Size	Medium Size	Large Size	Extra Large Size	Rendered Output
Parentheses	\bigl(	\Bigl(	\biggl(	\Biggl(
Brackets	\bigr[	\Bigr[	\biggr[	\Biggr[
Braces	\bigr\{	\Bigr\{	\biggr\{	\Biggr\{
Angle Brackets	\bigl\langle	\Bigl\langle	\biggl\langle	\Biggl\langle
Vertical Bars	\big|	\Big|	\bigg|	\Bigg|
Double Bars	\big|	\Big|	\bigg|	\Bigg|
Ceiling	\bigl\lceil	\Bigl\lceil	\biggl\lceil	\Biggl\lceil
Floor	\bigl\lfloor	\Bigl\lfloor	\biggl\lfloor	\Biggl\lfloor