Calculator Input
Enter finite elements separated by commas, semicolons, or new lines. Duplicate entries are removed automatically.
Example Data Table
| Scenario | Set A | Set B | Universal Set U | Main insight |
|---|---|---|---|---|
| Numbers | {1, 2, 3, 4} | {3, 4, 5} | {1, 2, 3, 4, 5, 6, 7} | Intersection is {3, 4}; complements are available. |
| Subjects | {Algebra, Geometry, Trigonometry} | {Geometry, Calculus} | {Algebra, Geometry, Trigonometry, Calculus, Statistics} | Use for overlap and curriculum coverage checks. |
| Letters | {a, b, c} | {c, d, e} | {a, b, c, d, e, f} | Useful for symbolic set relations and identities. |
Formula Used
This calculator works with finite sets and standard set algebra rules. It removes duplicates first, then applies each operation to the cleaned inputs.
Union
A ∪ B contains every distinct element found in A or B or both.
Intersection
A ∩ B contains only elements common to both sets.
Difference
A − B keeps elements in A that are not in B. Likewise, B − A reverses the direction.
Symmetric Difference
A △ B = (A − B) ∪ (B − A). It returns elements belonging to exactly one set.
Complement
Aᶜ = U − A and Bᶜ = U − B. Complements require a universal set U.
Cartesian Product
A × B lists ordered pairs. The cardinality formula is n(A × B) = n(A) × n(B).
Power Set Cardinality
For any finite set A, the number of subsets is |P(A)| = 2n(A).
Jaccard Index
J(A, B) = |A ∩ B| / |A ∪ B| when the union is not empty.
De Morgan Verification
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ, relative to U.
How to Use This Calculator
1. Enter Set A
Type the first finite set using commas, semicolons, or separate lines between elements.
2. Enter Set B
Add the second set using the same input style. Duplicate values are discarded automatically.
3. Add the Universal Set
Fill U when you need complements or De Morgan identity checks. Leave it blank otherwise.
4. Choose Sorting and Preview Options
Select automatic, numeric, lexicographic, or input-order sorting. Set a safe Cartesian preview limit.
5. Run the Calculation
Press the calculate button. Results appear above the form and below the page header.
6. Export Your Output
Use the CSV or PDF buttons to save the visible result summary for notes, classes, or assignments.
Frequently Asked Questions
1. What kind of sets can I enter?
You can enter any finite set of numbers, words, codes, labels, or symbols. The calculator reads each unique item as one element after trimming separators and duplicates.
2. Why are duplicate values removed?
A mathematical set stores distinct elements only. Repeated entries do not change the set, so the calculator automatically keeps one copy of each cleaned element.
3. When do I need the universal set?
You need a universal set when calculating complements or checking De Morgan identities. Without U, operations such as union and intersection still work normally.
4. Does order matter in the final set?
Order does not affect set membership, but display order helps readability. You can sort automatically, force numeric order, use lexicographic order, or preserve input order.
5. What is the Cartesian preview limit for?
Cartesian products grow quickly because every element in A pairs with every element in B. The preview limit keeps the page readable while still showing a representative sample.
6. Can I compare text values with different letter cases?
Yes. Keep case sensitivity enabled to treat A and a as different elements. Turn it off when you want case-insensitive matching.
7. What does the Jaccard index show?
The Jaccard index measures overlap strength between two sets. A value near 1 means strong similarity, while a value near 0 means little shared membership.
8. Can I use this for homework and teaching?
Yes. The calculator is useful for classroom demonstrations, worksheet checking, revision, and quick verification of finite set identities and relationships.