What Do Parentheses and Brackets Mean in Interval Notation?
The concept of interval notation is essential in mathematics and helps in solving real-world and exam-level problems efficiently. It provides a simple, compact way to express sets of real numbers on a number line, especially when dealing with inequalities, domains, and ranges.
Understanding Interval Notation
Interval notation is a system of writing subsets of real numbers using brackets and parentheses to show whether endpoints are included or excluded. This concept is widely used in inequalities, function domains and ranges, and in representing solution sets visually or algebraically. Interval notation offers an easy, standardized way to convey continuous intervals, making it a core skill for all students preparing for maths exams.
Types of Intervals in Interval Notation
There are three main types of intervals you will encounter in mathematics:
| Type | Notation | Endpoint Inclusion | Example |
|---|---|---|---|
| Open Interval | (a, b) | a and b excluded | (2,5) |
| Closed Interval | [a, b] | a and b included | [2,5] |
| Half-Open/Closed Interval | (a, b] or [a, b) | Includes only one endpoint | (2,5] or [2,5) |
This table shows how each form of interval notation specifies which numbers are included in your interval.
How to Write in Interval Notation
Follow these simple steps to write any range of numbers using interval notation:
1. Identify the smallest and largest values of the set.
2. Decide if endpoints should be included (use [ ] brackets) or excluded (use ( ) parentheses).
3. If the interval goes forever in a direction, use infinity (∞) or negative infinity (−∞) with a parenthesis.
4. Write the interval in the form: (a, b), [a, b], (a, b], [a, b), (−∞, b), (a, ∞), etc.
5. For sets made of separate intervals, combine with the union symbol: ∪
When reading interval notation in words:
(2,5) is "x is greater than 2 and less than 5" (2 < x < 5)
[2,5] is "x is greater than or equal to 2 and less than or equal to 5" (2 ≤ x ≤ 5)
Brackets, Parentheses, and Symbols in Interval Notation
It is very important to understand what the different brackets mean in interval notation.
| Symbol | Name | Usage |
|---|---|---|
| [ ] | Square Bracket | Endpoint included (“closed” interval) |
| ( ) | Parenthesis | Endpoint excluded (“open” interval) |
| ∪ | Union | Used to join two or more intervals |
| ∞, −∞ | Infinity symbols | Represent unbounded intervals |
Remember: Always use ( ) with infinity since infinity is not a real, reachable number.
Step-by-Step: Converting Inequalities to Interval Notation
Let’s see how to convert a given inequality into interval notation:
1. Write the inequality, for example: x ≥ 3
2. Look at the symbol (≥ means “greater than or equal to” so include the endpoint).
3. Since there is no upper bound, the right end is infinity: [3, ∞)
4. Because infinity is never included, always use a parenthesis with it.
Final answer: [3, ∞)
Worked Example – Solving Interval Notation Problems
Example 1: Write the set of all real numbers x such that −2 < x ≤ 5.
1. Recognize: The left side is open ("less than" only); the right is closed ("less than or equal to").2. Use '(' for −2 since it's not included, and ']' for 5 since it is included.
3. The final interval is (−2, 5]
Example 2: x ≤ 7
1. The left endpoint is unbounded; so begin with (−∞,2. The right endpoint is 7, and since ≤ includes 7, use ]
3. (−∞, 7]
Example 3: x > 0 or x < −3
1. This is two separate intervals.2. x > 0 is (0, ∞), and x < −3 is (−∞, −3)
3. Use union symbol: (−∞, −3) ∪ (0, ∞)
Interval Notation for Domain and Range
Interval notation is very useful for writing the domain and range of functions. For example, the function y = √x has a domain [0, ∞) because you can only take the square root of numbers 0 or greater. The range is also [0, ∞).
Interval Notation vs Set Notation
Interval notation is different from set notation.
| Set Notation | Equivalent Interval Notation |
|---|---|
| {x | 1 ≤ x < 4} | [1, 4) |
| {x | x > −2} | (−2, ∞) |
Both notations describe sets, but interval notation is shorter and easier to use for continuous numbers.
Practice Problems
- Write the interval notation for: 3 < x ≤ 8
- Express all real numbers less than 0 in interval notation.
- Convert x ≥ −2 and x < 5 to interval form.
- What is the interval notation for 0 ≤ x ≤ 10?
Common Mistakes to Avoid
- Mixing up parentheses and brackets – remember, ( ) means exclude, [ ] means include.
- Using [ ] with infinity – always use ( ) with infinity or −infinity.
- Writing endpoints in the wrong order – always write the smaller number first.
Real-World Applications
The concept of interval notation appears in statistics (age ranges, score bands), engineering (tolerances), programming, and functions' domains and ranges. Vedantu helps students see how maths applies beyond the classroom, such as expressing test marks between 30 and 80 as [30, 80].
Related Concepts and Further Reading
- Sets and their Representations
- Domain and Range Relations
- Linear Inequalities
- Relations and Functions
- Subsets and Supersets
- Intervals
- Numbers on the Number Line
We explored the idea of interval notation, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these topics and boost your exam performance.
