Build and test your universal set
Use commas, semicolons, pipes, or new lines as separators.
Example data table
This sample shows how a universal set supports complements, unions, and intersections.
| U | A | B | C | A ∪ B ∪ C | A ∩ B | Aᶜ |
|---|---|---|---|---|---|---|
| { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } | { 1, 2, 3, 4 } | { 3, 4, 5, 6 } | { 2, 6, 8 } | { 1, 2, 3, 4, 5, 6, 8 } | { 3, 4 } | { 5, 6, 7, 8, 9, 10 } |
Formula used
- Automatic universal set: U = A ∪ B ∪ C ∪ Extra
- Manual universal set: U = Entered U ∪ Extra
- Complement: Aᶜ = U \ A
- Union: A ∪ B = {x | x ∈ A or x ∈ B}
- Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
- Difference: A \ B = {x | x ∈ A and x ∉ B}
- Symmetric difference: A △ B = (A \ B) ∪ (B \ A)
- Power set size: |P(U)| = 2^|U|
How to use this calculator
- Select automatic mode to build U from A, B, C, and extra elements.
- Select manual mode when you already know the universal set.
- Enter items with commas, semicolons, pipes, or line breaks.
- Choose whether text case should stay unchanged or become lowercase.
- Pick a display order if you want sorted output.
- Optionally enter a probe element to test membership across all sets.
- Press the calculate button to show results above the form.
- Use the CSV and PDF buttons to export the generated summary.
FAQs
1. What is a universal set?
A universal set contains every element considered in the current problem. All complements are measured relative to this set, not relative to the entire mathematical universe.
2. What does automatic mode do?
Automatic mode builds U from all unique elements found in A, B, C, and the extra universe field. It is useful when the universe is implied by the entered subsets.
3. Why were some entered elements excluded?
In manual mode, any item that is not inside U cannot belong to a subset of U. The calculator flags those items and excludes them from complement-based operations.
4. What does the complement represent?
The complement of A is every element in U that does not appear in A. It is written as Aᶜ or U \ A.
5. Are duplicate values counted twice?
No. Sets store unique elements only. Repeated entries are merged into one value before the calculator performs any union, intersection, complement, or difference operation.
6. Can I use words instead of numbers?
Yes. You can enter labels such as red, blue, apple, or team-a. The calculator treats each cleaned entry as a distinct set element.
7. What is symmetric difference?
Symmetric difference shows elements that belong to exactly one of two sets. For A and B, it includes items in A only and items in B only.
8. Why is the power set size included?
The power set counts every possible subset of U. Its size equals 2^|U|, which helps when you want to understand the full subset space.