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# Lesson 6.3: Brackets and Braces

## Symbols

$\text{[ left bracket}$
⠈⠷

$\text{] right bracket}$
⠈⠾

$\mathbf{\left[}\phantom{\rule{.3em}{0ex}}\text{left bold bracket}$
⠸⠈⠷

$\mathbf{\right]}\phantom{\rule{.3em}{0ex}}\text{right bold bracket}$
⠸⠈⠾

$\text{{ left brace}$
⠨⠷

$\text{} right brace}$
⠨⠾

$\text{∅ null set}$
⠸⠴

## Brackets

In braille, brackets are variations of parentheses. The opening and closing brackets are formed by adding a dot four in the cell preceding: the basic opening symbol, dots one two three five six; and the basic closing symbol, dots two three four five six. Boldface brackets are represented by the boldface typeform indicator, dots four five six, preceding the bracket symbol, making it a three-cell configuration. Nemeth code brackets are always to be used, even if the brackets enclose literary expressions within a mathematical context.

• Opening, left, bracket, dot four dots one two three five six
• Closing, right, bracket, dot four dots two three four five six
• Boldface, left, bracket, dots four five six dot four dots one two three five six
• Boldface, right, bracket, dots four five six dot four dots two three four five six
• ## Use of Brackets

Brackets are used to group or enclose other signs of grouping to show a higher level of grouping. They also serve many of the same purposes as parentheses when they are used at this second level of grouping. In interval notation, they can be used to show that a range of values includes a certain value.

## Braces

Braille braces are two-cell symbols that are modifications of the opening and closing parentheses. The braces are formed by preceding the configurations used for parentheses with dots four six. The opening, left, brace is comprised of dots four six dots one two three five six; the closing, right, brace, dots four six dots two three four five six.

## Braces enclosing sets

Braces are often used to indicate set notation. Such a set may include numeric values, letters, or words. Frequently, a set is designated by a capital letter followed by an equals sign. Following the equals sign is a set of braces which enclose the elements of the set.

## Facing Braces-The Empty Set

The empty set in math indicates that no elements can be found that match the conditions for the set. It is often shown with two braces not containing any elements or with a zero and a slash or vertical line through it. When braces are used to represent the empty set, a space is inserted between the braces to indicate that the set does not have anything in it. This is not the same as an omitted item. For example, "the set of all cats which have been formally trained to guide blind persons" equals the empty set. This is because, as of this writing, there have not been any cats who have been formally trained to guide people who are blind. The space between the braces shows that nothing fulfills the requirement.

## Punctuation with Brackets and Braces

Brackets and braces are mathematical symbols, even when they are used to enclose literary material. They are to be punctuated as mathematical symbols. The punctuation indicator is used with all marks of punctuation other than the mathematical comma, hyphen, and dash.

## Special Rules for Brackets

Do not use the numeric indicator immediately following the opening bracket indicator except when the brackets are used to show a matrix. See rules for matrices.

Punctuate as mathematical symbols.

Brackets follow the general rules for signs of grouping.

## Special Rules for Braces

In braille, braces should appear in the same position as they do in print.

The following may not be used in direct contact with a sign of grouping, such as the brace: one-cell whole-word alphabet contractions, lower-cell whole word contractions, or any of the whole or part-word contractions, such as and, for, of, the, or with.

Use or non-use of the English letter indicator depends upon whether or not it would be required if the braces were not present. In lists of items separated by commas, it is not required.

The numeric indicator is not used with numerals in an enclosed list or when numerals are in contact with both signs of grouping.

Punctuate braces as mathematical symbols.

### Example 1

$\left[7+\left(10÷2\right)\right]$
⠈⠷⠶⠬⠷⠂⠴⠨⠌⠆⠾⠈⠾

### Example 2

Bold face brackets

$\mathbf{\left[}2.1\mathbf{\right]}=2$
⠸⠈⠷⠆⠨⠂⠸⠈⠾⠀⠨⠅⠀⠼⠆

### Example 3

$\left[-5,\phantom{\rule{.3em}{0ex}}2\right)$
⠈⠷⠤⠢⠠⠀⠆⠾

### Example 4

$\left[-5,\phantom{\rule{.3em}{0ex}}2\right]$
⠈⠷⠤⠢⠠⠀⠆⠈⠾

### Example 5

$B=\left\{-1,\phantom{\rule{.3em}{0ex}}2,\phantom{\rule{.3em}{0ex}}2.4\right\}$
⠠⠃⠀⠨⠅⠀⠨⠷⠤⠂⠠⠀⠆⠠⠀⠆⠨⠲⠨⠾

### Example 6

The vertical bar is a sign of comparison and should be brailled according to those rules.

$A=\left\{x|x=3y\right\}$
⠠⠁⠀⠨⠅⠀⠨⠷⠭⠳⠭⠀⠨⠅⠀⠼⠒⠽⠨⠾

### Example 7

$\left\{5,\phantom{\rule{.3em}{0ex}}-1,\phantom{\rule{.3em}{0ex}}2,\phantom{\rule{.3em}{0ex}}r,\phantom{\rule{.3em}{0ex}}p,\phantom{\rule{.3em}{0ex}}B\right\}$
⠨⠷⠢⠠⠀⠤⠂⠠⠀⠆⠠⠀⠗⠠⠀⠏⠠⠀⠠⠃⠨⠾

### Example 8

$\left\{0\right\}$
⠨⠷⠴⠨⠾

### Example 9

$\left\{\right\}\phantom{\rule{.3em}{0ex}}\text{or}\phantom{\rule{.3em}{0ex}}\varnothing$
⠨⠷⠀⠨⠾⠀⠕⠗⠀⠸⠴