Interval Union & Intersection (Math Guide)
The union of two intervals contains every value in either interval; their intersection contains only values shared by both. Read ∪ as “or” and ∩ as “and.” To find either result, mark the intervals on a number line, compare their endpoints, and check whether each boundary is included or excluded.
Interval notation can look like a small detail, but one bracket or parenthesis can change the answer. A number line makes the difference visible: the union traces all covered parts, while the intersection keeps only the shared part. I often find that drawing the endpoints first helps prevent mistakes, especially when intervals meet at a single point.
Interval Operations: Purpose, Notation, and Prerequisite Skills
An interval is a set of real numbers between two bounds, with rules about whether its endpoints belong to the set. Union and intersection are ways to combine sets. Before using them, identify each interval’s bounds and endpoint symbols, then connect those symbols to inequalities and points on a number line.
An interval can include or exclude either endpoint. In the interval ([1,4)), for example, 1 is included and 4 is excluded. The same set can be written as (1\leq x<4). A square bracket means “included”; a parenthesis means “not included.”
An interval may also have no finite bound on one side. For example, ((-\infty,2]) contains every real number less than or equal to 2. Infinity is never an endpoint value, so it always uses a parenthesis.
| Notation | Endpoint meaning | Inequality example |
|---|---|---|
| ([a,b]) | Includes (a) and (b) | (a\leq x\leq b) |
| ((a,b)) | Excludes (a) and (b) | (a<x<b) |
| ([a,b)) | Includes (a), excludes (b) | (a\leq x<b) |
| ((a,b]) | Excludes (a), includes (b) | (a<x\leq b) |
| ((-\infty,b]) | No lower bound; includes (b) | (x\leq b) |
The set of real numbers is often written (\mathbb R). The empty set, written (\varnothing), has no elements. These symbols are useful when intervals do not overlap.
Union and Intersection: Rules, Endpoint Requirements, and Worked Examples
The union (A\cup B) contains values in (A), in (B), or in both. The intersection (A\cap B) contains only values that belong to both. To calculate either operation, compare the intervals’ positions and then apply the endpoint rules at every boundary.
How to find a union
For a union, combine the regions covered by either interval. If the intervals overlap, the result is usually one continuous interval. If they have a gap, keep them separate.
For example, let (A=[1,4)) and (B=(3,6]). The intervals overlap between 3 and 4, so together they cover every number from 1 through 6:
[ [1,4)\cup(3,6]=[1,6]. ]
The result includes 1 because (A) includes it, and includes 6 because (B) includes it.
When two intervals meet at the same endpoint, check whether at least one interval includes that point. For instance:
[ [1,3)\cup[3,5]=[1,5]. ]
The point 3 belongs to the second interval, so there is no missing point between the two regions.
But if neither interval includes the shared point, the union has a hole:
[ [1,3)\cup(3,5]=[1,3)\cup(3,5]. ]
Here, 3 belongs to neither interval. The answer cannot be written as ([1,5]), because that would wrongly include 3.
How to find an intersection
For an intersection, keep only the values that both intervals contain. The overlap begins at the larger lower bound and ends at the smaller upper bound. Then check whether those boundary values are included in both intervals.
Using (A=[1,4)) and (B=(3,6]), the shared values are greater than 3 and less than 4:
[ [1,4)\cap(3,6]=(3,4). ]
Neither 3 nor 4 belongs to both intervals. In fact, 3 is excluded by (B), and 4 is excluded by (A).
If intervals touch at just one point, their intersection may be that single point, or it may be empty:
[ [1,3]\cap[3,5]={3}, ] [ [1,3)\cap[3,5]=\varnothing. ]
In the first example, both intervals include 3. In the second, the first interval stops just before 3, so there is no shared value.
| Intervals | Union | Intersection | Why |
|---|---|---|---|
| ([1,4)), ((3,6]) | ([1,6]) | ((3,4)) | They overlap from just above 3 to just below 4. |
| ([1,3)), ([3,5]) | ([1,5]) | (\varnothing) | The second interval includes 3; the first does not. |
| ([1,3]), ([3,5]) | ([1,5]) | ({3}) | Both include the shared endpoint. |
| ([1,2]), ([4,6)) | ([1,2]\cup[4,6)) | (\varnothing) | There is a gap between 2 and 4. |
A set containing one number can be written with braces, as in ({3}). It is not an interval with a stretch of numbers; it is a single shared value.
Number-Line Methods, Practice Sequence, and Common Errors
A number line helps you see which regions belong to each interval and where they overlap. Use it to check your result, not to replace the notation rules. Mark open and closed endpoints carefully, then test important boundary values before writing the final union or intersection.
Use this sequence for a reliable solution:
- Read each interval. Write down its lower and upper bounds. Note which finite endpoints are included.
- Translate the operation. For (\cup), include values in either set. For (\cap), keep values in both.
- Mark both intervals. Use an open circle for an excluded endpoint and a filled circle for an included endpoint.
- Identify the result. For a union, follow all covered regions. For an intersection, mark only the overlap.
- Check the boundaries. Test each shared endpoint and any point where the regions begin or end.
- Write the answer. Use interval notation, or use a union of intervals if the result has a gap.
For example, compare ((-2,3]) and ([1,5)). Their union covers every value greater than (-2) and less than 5, so it is ((-2,5)). Their intersection runs from 1 through 3, with both endpoints included, so it is ([1,3]).
A simple sketch might look like this:
A: -2 ○────────● 3
B: 1 ●────────○ 5
The shared region starts at 1 and ends at 3. Both are filled points, so the intersection is ([1,3]).
Common errors to watch for
- Reversing the operations: Union is not just the overlap. Intersection is not everything covered by either interval.
- Including a point because the intervals touch: Touching does not guarantee a shared point. Check both endpoint symbols.
- Copying an endpoint symbol without checking: The lower or upper bound of the answer may come from one interval, or from both.
- Writing a gap as one interval: If some values between the regions are missing, use two intervals joined by (\cup).
- Using a parenthesis at infinity incorrectly: Always write ((-\infty,a)) or ((a,\infty)), never a square bracket at infinity.
A useful boundary test is to ask, “Does this exact number belong to both sets?” for an intersection, or “Does it belong to at least one?” for a union. This catches many endpoint errors before they reach the final answer.
Applications of Interval Operations in Inequalities and Further Mathematics
Intervals often describe the solutions to inequalities. When two conditions must both hold, find an intersection; when either condition is enough, find a union. Converting between interval notation and inequalities makes the logic clear and helps you check whether your combined solution includes the right values.
Suppose a value (x) must satisfy (x\geq 2) and (x<7). The two conditions hold together for (2\leq x<7), or ([2,7)). This is an intersection because both conditions must be true.
Now suppose (x<-1) or (x\geq 3). Either condition is enough, so the solution is a union:
[ (-\infty,-1)\cup[3,\infty). ]
The gap from (-1) through values less than 3 is not part of the answer. The endpoint (-1) is excluded, while 3 is included.
The words in a problem often signal the operation:
| Wording in a problem | Set operation | What to keep |
|---|---|---|
| “and,” “both,” “at the same time” | Intersection, (\cap) | Values satisfying every condition |
| “or,” “either,” “at least one” | Union, (\cup) | Values satisfying one or both conditions |
For inequalities with several conditions, solve each condition first, then combine the solution sets. If a problem says (x>0) and (x\leq 4), the answer is ((0,\infty)\cap(-\infty,4]= (0,4]).
A final check is simple: choose one value from the proposed solution and test it in the original conditions. Also test a nearby value just outside the answer. If the first fails the stated rule, or the second meets it, review the endpoint or operation.
Conclusion: A Quick Check Before You Submit
A correct interval answer reflects both the covered values and the endpoint rules. Before finishing, confirm whether the task asks for “either” or “both,” inspect every boundary, and compare the result with a number line or the original inequalities. These checks make your notation precise and easy to read.
Keep the main distinction in view: union gathers all values from either set, while intersection keeps only shared values. A bracket or parenthesis records whether an endpoint belongs; it is part of the answer, not decoration.
When in doubt, check one boundary value at a time. For an intersection, the value must be in both intervals. For a union, it must be in at least one. Then confirm that any gap remains visible in the final notation.
Frequently Asked Questions
These short answers review the most common questions about interval unions, intersections, and endpoints. Use them as a quick check after solving a problem. When a case depends on a boundary, the endpoint symbols in the original intervals decide whether that value is included.
What does the union of two intervals mean?
The union contains every number that belongs to at least one of the intervals. It includes numbers in either interval and numbers shared by both.
What does the intersection of two intervals mean?
The intersection contains only numbers that belong to both intervals. If the intervals share no values, their intersection is (\varnothing).
Is union “and” or “or”?
Union matches “or.” A value belongs to the union if it is in the first interval, the second interval, or both.
Is intersection “and” or “or”?
Intersection matches “and.” A value must belong to both intervals to be in their intersection.
What happens when two intervals touch at one endpoint?
For the intersection, the shared endpoint is included only if both intervals include it. For the union, that point is covered if at least one interval includes it.
Why can ([1,3)\cap[3,5]) be empty?
The first interval excludes 3, and the second includes 3. No number belongs to both intervals, so their intersection is (\varnothing).
When should I use two intervals in a union?
Use two intervals when there is a gap between the covered regions, or when a shared point is excluded by both intervals. For example, ([1,3)\cup(3,5]) leaves out 3.
Can an intersection contain only one number?
Yes. If two intervals share exactly one included endpoint, the intersection is a single number, such as ([1,3]\cap[3,5]={3}).
How do I handle infinity in interval notation?
Use a parenthesis next to infinity because infinity is not a real number that can be included. For example, numbers at most 4 are written ((-\infty,4]).
How can I check my answer?
Mark both intervals on a number line and inspect the endpoints. Then test a boundary value and a value from each region to confirm that the result matches “either” for union or “both” for intersection.
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
