Types of Intervals
Imagine you have a number line. An interval is simply a connected chunk of that line — all the real numbers that lie between two endpoints. The question is: do you include the endpoints themselves, or not? That choice gives you the different types of intervals.
The Intuition
Suppose you're told "pick any number between 2 and 5". That's vague. Do you mean:
- Any number from 2 to 5, including 2 and 5 themselves? (So 2, 5, and everything in between are allowed.)
- Any number from 2 to 5, excluding 2 and 5? (So 2.0001 is fine, but 2 itself is not.)
- Include one endpoint but not the other? (Like "from 2 up to, but not including, 5".)
Each of these is a different type of interval. The notation and names are designed to make this crystal clear.
The Precise Statement
An interval is a set of real numbers. For two real numbers a and b with a<b, there are four standard types:
Open: (a,b)={x∈R∣a<x<b}
Closed: [a,b]={x∈R∣a≤x≤b}
Half-open (left): (a,b]={x∈R∣a<x≤b}
Half-open (right): [a,b)={x∈R∣a≤x<b}
The round bracket ( or ) means "this endpoint is not included". The square bracket [ or ] means "this endpoint is included".
| Name | Notation | Includes a? | Includes b? | Example numbers |
|---|
| Open | (a,b) | No | No | 2.1,3,4.999 but not 2 or 5 |
| Closed | [a,b] | Yes | Yes | 2,5, and everything in between |
| Left-open | (a,b] | No | Yes | 2.001 up to and including 5 |
| Right-open | [a,b) | Yes | No | 2 up to but not including 5 |