The primary difference between open and closed brackets lies in whether the endpoints of an interval are included or excluded.
Open vs. Closed Brackets: A Detailed Comparison
Open and closed brackets are mathematical notations used to define intervals on the number line. Understanding the difference between them is crucial in various mathematical contexts, including set theory, calculus, and analysis.
Open Brackets
- Definition: Open brackets, denoted by parentheses
( ), indicate that the endpoint values are not included in the interval. - Representation: An open interval from a to b is written as
(a, b). - Meaning: This represents all numbers between a and b, excluding a and b themselves.
Closed Brackets
- Definition: Closed brackets, denoted by square brackets
[ ], indicate that the endpoint values are included in the interval. - Representation: A closed interval from a to b is written as
[a, b]. - Meaning: This represents all numbers between a and b, including a and b.
Summary Table
| Feature | Open Bracket ( ) |
Closed Bracket [ ] |
|---|---|---|
| Endpoints | Excluded | Included |
| Interval Type | Open Interval | Closed Interval |
| Representation | (a, b) |
[a, b] |
Examples
(2, 5): Represents all numbers between 2 and 5, but not including 2 and 5.[2, 5]: Represents all numbers between 2 and 5, including 2 and 5.(2, 5]: Represents all numbers greater than 2 and less than or equal to 5. 2 is excluded and 5 is included.[2, 5): Represents all numbers greater than or equal to 2 and less than 5. 2 is included and 5 is excluded.
As the provided information states: In the open intervals, the set of numbers which represent the endpoints are not included, and in closed intervals, the set of numbers which represent the endpoints are included. The open interval is represented using parenthesis - ( ), and the closed interval is represented using box brackets - [ ].
