Union vs Intersection of Intervals Explained
6 min read
Intervals are one of the most practical ways to describe continuous ranges of numbers. They appear in algebra, calculus, statistics, programming, scheduling, measurement, and data analysis. When working with intervals, two operations are especially important: the union and the intersection. Understanding the difference between them helps you reason clearly about what values are included, excluded, shared, or combined.
TLDR: The union of intervals includes every number that belongs to at least one of the intervals. The intersection includes only the numbers that belong to all intervals being compared. In simple terms, union means “combine,” while intersection means “overlap.” The key is to pay close attention to endpoints, open and closed brackets, and whether the intervals actually touch or overlap.
What Is an Interval?
An interval is a set of numbers between two endpoints. For example, the interval from 2 to 7 contains all numbers between 2 and 7, such as 3, 4.5, 6.99, and so on. Intervals are useful because they describe continuous ranges rather than isolated values.
Intervals are usually written using brackets and parentheses:
- [a, b] means all numbers from a to b, including both endpoints.
- (a, b) means all numbers between a and b, excluding both endpoints.
- [a, b) means a is included, but b is excluded.
- (a, b] means a is excluded, but b is included.
For instance, [2, 5] includes 2 and 5, while (2, 5) does not include either endpoint. This distinction becomes very important when finding unions and intersections.
The Union of Intervals
The union of two or more intervals is the set of all numbers that appear in at least one of the intervals. In mathematical notation, the union symbol is ∪. If a number is in the first interval, the second interval, or both, it is included in the union.
For example, consider the intervals:
A = [1, 4] and B = [3, 6]
The union is:
A ∪ B = [1, 6]
Why? Because the first interval covers everything from 1 to 4, and the second interval covers everything from 3 to 6. Together, they cover the entire range from 1 to 6. Since the intervals overlap between 3 and 4, there is no gap between them.
Union is often understood as a combining operation. It answers the question: “What values are included if we take everything from both intervals?”
The Intersection of Intervals
The intersection of two or more intervals is the set of all numbers that appear in every interval being considered. In mathematical notation, the intersection symbol is ∩. A number must be common to all intervals in order to be included in the intersection.
Using the same example:
A = [1, 4] and B = [3, 6]
The intersection is:
A ∩ B = [3, 4]
This is because the numbers from 3 to 4 are contained in both intervals. Values less than 3 are only in A, and values greater than 4 are only in B. Therefore, the shared part is exactly [3, 4].
Intersection is best understood as an overlap operation. It answers the question: “What values do these intervals have in common?”
Union vs Intersection: The Core Difference
The difference between union and intersection can be summarized clearly:
- Union includes values that belong to one interval or the other, or both.
- Intersection includes only values that belong to both intervals at the same time.
If intervals are regions on a number line, the union is the total shaded area covered by all regions. The intersection is only the part where the shaded regions overlap.
For example:
[2, 8] ∪ [5, 10] = [2, 10]
[2, 8] ∩ [5, 10] = [5, 8]
The union stretches from the earliest included point to the latest included point because the intervals overlap. The intersection is limited to the shared portion.
When Intervals Do Not Overlap
Not all intervals overlap. Suppose:
A = [1, 3] and B = [5, 7]
These intervals are separated by a gap between 3 and 5. Therefore, their union is not a single continuous interval:
A ∪ B = [1, 3] ∪ [5, 7]
The union includes both separate intervals. However, the intersection is empty because there is no number that belongs to both intervals:
A ∩ B = ∅
The symbol ∅ means the empty set, which means there are no shared values.
This is a common point of confusion. A union does not always produce one continuous interval. If the intervals are disconnected, the union must be written as separate pieces.
When Intervals Touch at One Point
Intervals may not overlap over a range but may still touch at a single endpoint. In that case, the brackets determine whether the endpoint is included.
Consider:
A = [1, 4] and B = [4, 7]
Both intervals include the number 4. Therefore:
A ∪ B = [1, 7]
A ∩ B = {4}
The intersection is the single number 4, usually written as {4}, because that is the only value common to both intervals.
Now compare this with:
A = [1, 4) and B = (4, 7]
Here, neither interval includes 4. The first interval stops just before 4, and the second starts just after 4. Therefore:
A ∪ B = [1, 4) ∪ (4, 7]
A ∩ B = ∅
Although the intervals appear to meet at 4, the number 4 itself is excluded from both. As a result, there is a gap at exactly 4.
Open and Closed Endpoints Matter
The most careful part of working with intervals is deciding whether endpoints are included. A square bracket [ ] means the endpoint is included. A parenthesis ( ) means the endpoint is not included.
For example:
A = (2, 6] and B = [6, 9)
The number 6 is included in A because A ends with a square bracket. It is also included in B because B begins with a square bracket. Therefore:
A ∩ B = {6}
And the union is:
A ∪ B = (2, 9)
Notice that 2 is excluded and 9 is excluded. The union covers everything between 2 and 9, but not the endpoints 2 and 9 themselves.
Now consider:
A = (2, 6) and B = (6, 9)
In this case, 6 is excluded from both intervals. So:
A ∩ B = ∅
A ∪ B = (2, 6) ∪ (6, 9)
The single missing point at 6 prevents the union from being written as (2, 9).
Intervals with Infinity
Intervals can extend indefinitely using infinity symbols. For example:
(−∞, 5] means all numbers less than or equal to 5.
[3, ∞) means all numbers greater than or equal to 3.
Infinity is never included as an endpoint because it is not a real number. Therefore, infinity always uses a parenthesis, not a square bracket.
Now consider:
A = (−∞, 5] and B = [3, ∞)
The union is:
A ∪ B = (−∞, ∞)
Together, these intervals cover every real number. The intersection is:
A ∩ B = [3, 5]
Only the numbers from 3 to 5 are common to both intervals.
A Reliable Method for Solving Interval Problems
To avoid mistakes, use a structured approach. The following method works well for most union and intersection problems:
- Draw or imagine a number line. Place all relevant endpoints in order.
- Mark open and closed endpoints. Use open circles for excluded endpoints and filled circles for included endpoints.
- Shade each interval. This makes overlap and gaps easier to see.
- For union, collect all shaded regions. Include everything covered by at least one interval.
- For intersection, keep only the regions shaded by every interval. Exclude anything not shared.
- Check endpoints carefully. Decide whether each boundary point is included or excluded.
This approach is especially helpful when intervals have mixed brackets, touch at endpoints, or extend to infinity.
Common Mistakes to Avoid
Several errors occur frequently when students and professionals work with intervals. The most common include:
- Combining separated intervals incorrectly. For example, writing [1, 3] ∪ [5, 7] as [1, 7] is wrong because there is a gap from 3 to 5.
- Ignoring open endpoints. The intervals [1, 4) and [4, 6] do not share the number 4 because the first interval excludes it.
- Confusing union with intersection. Union is broader; intersection is narrower.
- Including infinity as an endpoint. Expressions like [−∞, 5] are incorrect because infinity cannot be included.
Real-World Interpretation
Union and intersection are not just abstract mathematical ideas. They are useful in real situations. Suppose one machine operates safely between 10 and 25 degrees, while another operates safely between 20 and 35 degrees. The intersection, [20, 25], tells you the temperature range where both machines can safely operate together. The union, [10, 35], tells you the full range covered by at least one machine.
In scheduling, if one person is available from 9:00 to 12:00 and another from 11:00 to 14:00, the intersection is 11:00 to 12:00, the time when both are available. The union is 9:00 to 14:00 if you are describing the total time during which at least one person is available.
Final Thoughts
The distinction between union and intersection is fundamental. The union of intervals gathers all values included by any interval, while the intersection keeps only the values shared by all intervals. In practice, the correct answer depends not only on where intervals begin and end, but also on whether those endpoints are included.
A dependable way to remember the difference is this: union expands, while intersection narrows. Union gives the full coverage; intersection gives the common ground. Once you understand that principle and handle endpoints carefully, interval problems become much more predictable and precise.
