# 9.1 Set Notations: Complements, Subsets, Union, Intersection

## Complement to a Set

When working with sets, they are typically named using capital letters. Sets are frequently parts of larger sets. The complement, or items that are not in a specified set, are often written with the same capital letter followed by what looks like an apostrophe in print. Thus, if you have a set A, the complement of A would be written as A′. In braille this is a letter followed by a dot three. The apostrophe, dot three, is also used to represent feet or prime symbol.

## Review

In previous lessons, it was noted that the empty set may be represented by an empty set of braces facing each other, {}. It is also known as the null set or ∅.

## Element, or a member of, a set

An item within a set is referred to as an element of or a member of the set. The braille symbol for element of a set is dot four dots one five. To remember the symbol you may wish to associate it with the letter e for element, since the second cell has the same braille configuration as the letter e. The print symbol resembles the Greek letter epsilon and may be referred to as such in some texts. However, as a mathematical symbol, epsilon is used to represent an exceptionally small amount. In braille, the symbol is always represented by dot four dots one five, no matter how it may appear in print. The symbol for "not an element of" is represented in braille by dots three four dot four dots one five.

## Subset, also called proper subset or inclusion

Subsets are entire sets that are included within another set. The braille configuration of the symbol for subset, proper subset or inclusion, is dots four five six dot five dots one three.

## Subset or equivalent to

The symbol for a subset of or equal to is the subset symbol with a horizontal line displayed beneath it. The symbol often appears in print with only one line of the equals sign and occasionally with the full equals sign. The braille configuration for subset or equivalent to, using a single horizontal line, is dots four five six dot five dots one three dots one five six. The braille configuration for subset or equivalent to, using a double horizontal line, is dots four five six dot five dots one three dots four six dots one three.

## Proper superset, contains or implies

Dots four five six dots four six dot two is the braille configuration for the symbol for proper superset, also known as reverse inclusion.

Any of these signs can be negated using dots three four.

### Example 1

The complement to A is A′.

⠠⠞⠓⠑⠀⠉⠕⠍⠏⠇⠑⠍⠑⠝⠞⠀⠞⠕⠀⠸⠩⠀⠠⠁⠀⠠⠄⠊⠎⠀⠠⠁⠄⠀⠸⠱⠲

### Example 2

If A={1, 2, 4}, what is A′?

⠠⠊⠋⠀⠸⠩⠀⠠⠁⠀⠨⠅⠀⠨⠷⠂⠠⠀⠆⠠⠀⠲⠨⠾⠠⠀⠸⠱⠀⠺⠓⠁⠞⠀⠊⠎⠀⠸⠩⠀⠠⠁⠄⠀⠸⠱⠦

### Example 3

Three is an element of A.

⠼⠒⠀⠈⠑⠀⠠⠁

### Example 4

Three is not an element of A.

⠼⠒⠀⠌⠈⠑⠀⠠⠁

### Example 5

A is a subset of B

⠠⠁⠀⠸⠐⠅⠀⠠⠃

### Example 6

A is a subset of or equal to B (with one line of the equal)

⠠⠁⠀⠸⠐⠅⠱⠀⠠⠃

### Example 7

A is a subset of or equal to B (with both lines of the equal)

⠠⠁⠀⠸⠐⠅⠨⠅⠀⠠⠃

### Example 8

B is a superset of C

⠠⠃⠀⠸⠨⠂⠀⠠⠉

### Example 9

B is a superset or equal to C

⠠⠃⠀⠸⠨⠂⠱⠀⠠⠉