Types of intervals
An interval is a set of real numbers that sit between two ends, with no gaps. The two ends are called endpoints. Each end can be in the interval or out of it.
| Name | Inequality | Interval | Ends |
|---|---|---|---|
| Closed | a โค x โค b | [a, b] | both in |
| Open | a < x < b | (a, b) | both out |
| Half-open | a โค x < b | [a, b) | a in, b out |
| Half-open | a < x โค b | (a, b] | a out, b in |
Some intervals have no end on one side. They are unbounded: [a, โ) means x โฅ a, (a, โ) means x > a, (โโ, b] means x โค b, (โโ, b) means x < b, and (โโ, โ) is the whole line โ. The symbol โ ("infinity") only says "it never stops". It is not a number, so it always gets a round bracket.
Drawing intervals on the number line
To draw an interval, follow four small steps.
- Draw a number line and mark the two ends.
- At each end put a filled dot if the end is in, or a hollow dot if it is out.
- Colour the line between the ends.
- If the interval is unbounded, end the colour with an arrow.
Example: x > โ1 is (โ1, โ). Hollow dot at โ1, colour to the right, arrow at the end.
Integers inside an interval: count only the whole numbers that are in. For (โ2, 3] they are โ1, 0, 1, 2, 3, which makes 5 numbers. But the interval itself holds infinitely many numbers, such as 0.5 and 2.75.
Intersection and union of intervals
Take two intervals A and B.
Intersection A โฉ B: the numbers that are in A and in B. On the line, it is the overlap. Its left end is the bigger of the two left ends. Its right end is the smaller of the two right ends. If an end number is shared, it is in only when it is in both intervals.
Union A โช B: the numbers that are in A or in B (or both). On the line, it is all the colour. If A and B overlap or touch, the union is one interval. If there is a gap, the union is written as two pieces joined by โช.
Example: A = [โ3, 1], B = (โ1, 4]. Left end of overlap: the bigger of โ3 and โ1 is โ1, which B leaves out, so (. Right end: the smaller of 1 and 4 is 1, which A includes, so ]. A โฉ B = (โ1, 1]. A โช B = [โ3, 4].
If the intersection is empty we write โ . For A = [0, 1] and B = [2, 3] there is no overlap, so A โฉ B = โ and A โช B = [0, 1] โช [2, 3].
Try it: the number-line game
Play with a friend. One person says an interval, like (โ1, 4]. The other draws it, then names one number that is in it and one number that is just outside. Now change roles with two intervals and find A โฉ B and A โช B. The 3D in the last story step lets you check your answers.
Key formulas and definitions
- [a, b] = {x : a โค x โค b} closed; (a, b) = {x : a < x < b} open
- [a, b) and (a, b] are half-open
- โ and โโ always get a round bracket
- A โฉ B: numbers in both. A โช B: numbers in at least one
- Left end of A โฉ B = bigger left end; right end = smaller right end
Worked examples
1. Write โ1 โค x < 4 as an interval and say how to draw it.
It is [โ1, 4). Filled dot at โ1 (in), hollow dot at 4 (out), colour between them.
2. Write x > 5 as an interval.
x is bigger than 5 and never stops: (5, โ). 5 is out and โ always has a round bracket.
3. A = [โ2, 3] and B = (1, 6]. Find A โฉ B and A โช B.
Overlap starts at the bigger left end, 1. B leaves 1 out, so (. It ends at the smaller right end, 3. A has 3 in, so ]. A โฉ B = (1, 3]. The colour goes from โ2 to 6: left end โ2 is in A, right end 6 is in B. A โช B = [โ2, 6].
4. A = [0, 2] and B = (2, 5). Find A โฉ B and A โช B.
The number 2 is in A but not in B, so no number is in both: A โฉ B = โ . But A ends at 2 (included) and B starts just after 2, so there is no gap. A โช B = [0, 5).
5. A = (โโ, 3) and B = [โ1, โ). Find A โฉ B and A โช B.
Overlap: from โ1 (in B) to 3 (out of A): [โ1, 3). Together they cover every number: A โช B = (โโ, โ) = โ.
6. How many integers are in (โ2, 3]? List them.
โ2 is out, 3 is in. The integers are โ1, 0, 1, 2, 3. That is 5 integers.
Common mistakes
- Using a square bracket next to โ. Infinity is not a number, so it needs a round bracket: [1, โ).
- Mixing up the brackets: [ ] means included, ( ) means left out.
- Writing the bigger number first, like (5, 2). The smaller end always comes first.
- Joining two separate pieces as one interval. [0, 1] โช [2, 3] has a gap, so it is not [0, 3].