﻿ Interval and List Notation

# Interval and List Notation

The concepts for this lesson are summarized in Introduction to Sets.

Recall:

Braces look like $\,\{\;\;\}\,$ and are used for list notation.

Parentheses look like $\,(\;\;)\,$ and are used in interval notation when an endpoint IS NOT included.

Brackets look like $\,[\;\;\;]\,$ and are used in interval notation when an endpoint IS included.

## Examples

Question: Is $\,2\,$ in the set $\,(2,3)\,$?
Solution: No. The parenthesis next to the $\,2\,$ indicates that $\,2\,$ is not included.
Question: Is $\,2\,$ in the set $\,[2,3)\,$?
Solution: Yes. The bracket next to the $\,2\,$ indicates that $\,2\,$ is included.
Question: Is $\,2.5\,$ in the set $\,(2,3)\,$?
Solution: Yes. The interval $\,(2,3)\,$ contains all real numbers between $\,2\,$ and $\,3\,,$ but does not include either endpoint.
Question: Is $\,2\,$ in the set $\,\{2,3\}\,$?
Solution: Yes. This set has two members: the number $\,2\,,$ and the number $\,3\,.$ The braces indicate that list notation is being used here.
Question: Is $\,100\,$ in the set $\,\{1,2,3,\ldots\}\,$?
Solution: Yes. The ‘$\,\ldots\,$’ indicates that the established pattern continues ad infinitum. This set contains all positive integers.
Question: Is $\,100.5\,$ in the set $\,\{1,2,3,\ldots\}\,$?
Solution: No. The number $\,100.5\,$ is not an integer.
Question: Is $\,100.5\,$ in the set $\,(2,\infty)\,$?
Solution: Yes. This set contains all real numbers strictly greater than $\,2\,.$
Question: Is $\,2\,$ in the set $\,(-\infty,2)\,$?
Solution: No. The parenthesis next to the $\,2\,$ indicates that $\,2\,$ is not included.
Question: Is $\,2\,$ in the set $\,(-\infty,2]\,$?
Solution: Yes. The bracket next to the $\,2\,$ indicates that $\,2\,$ is included.
Question: Is $\,1.9999\,$ in the set $\,(-\infty,2)\,$?
Solution: Yes. This interval contains all real numbers less than $\,2\,.$