Family of Sets Calculator

Enter your family of sets

Use one set per line, such as A = {1,2,3} or B: red, blue. Unnamed lines are labeled automatically.

Family input

Universe U

Focus element

Treat uppercase and lowercase as the same

This calculator finds union, intersection, subset relations, disjointness, laminar structure, closure properties, complements, and element frequencies.

Example data table

Set	Elements	Cardinality	Notes
A	{1, 2, 3, 4}	4	Base set in the family
B	{2, 4, 6}	3	Shares elements with A and C
C	{1, 2, 4, 6, 8}	5	Largest set in this example
D	{2, 4}	2	Proper subset of A, B, and C

Formula used

Family size: |F| = number of sets listed in the family.

Union: ∪F = A₁ ∪ A₂ ∪ ... ∪ A_n. It contains every distinct element appearing anywhere.

Intersection: ∩F = A₁ ∩ A₂ ∩ ... ∩ A_n. It contains only elements common to every set.

Cardinality: |A| = number of distinct elements in set A.

Coverage ratio: Coverage = ( |∪F| / |U| ) × 100, when a universe U is supplied.

Subset test: A ⊆ B when every element of A is also an element of B.

Pairwise disjoint: A_i ∩ A_j = ∅ for every different pair of sets.

Laminar family: Any two sets are either disjoint or one is contained in the other.

Jaccard similarity: J(A,B) = |A ∩ B| / |A ∪ B|. The heatmap shows this value as a percentage.

Complement: A^c = U \ A, computed only when set A stays inside the chosen universe U.

Closure checks: The family is union-closed, intersection-closed, or complement-closed only when the produced set already belongs to the same family.

How to use this calculator

Enter one set per line. You may write A = {1,2,3}, B: red, blue, or simply {4,5,6}.
Optionally enter a universe if you want complements and coverage percentages.
Add a focus element to check which sets contain that item.
Choose case normalization if uppercase and lowercase items should be treated as identical.
Press Calculate Family Properties to show the result block above the form.
Review the summary table, set metrics, frequency table, matrices, and graphs.
Use the CSV button to export tables into spreadsheet-friendly text.
Use the PDF button to save a clean report of the calculated family properties.

Frequently asked questions

1. What is a family of sets?

A family of sets is a collection of sets treated as one mathematical object. The calculator studies the whole collection, not only each set individually.

2. Do I need to name every set?

No. Unnamed lines are accepted. The calculator automatically assigns labels like S1, S2, and S3 when no explicit set name is given.

3. Why should I enter a universe?

A universe lets the calculator compute complements and coverage ratios. Without it, those measures are mathematically undefined or incomplete for many problems.

4. What does union-closed mean?

A family is union-closed when the union of every pair of member sets is also a member of the same family. The calculator lists missing unions when closure fails.

5. What does laminar family mean?

A laminar family allows only two possibilities for any pair: the sets are disjoint, or one contains the other. Partial overlaps break laminar structure.

6. How is the heatmap value calculated?

The heatmap uses Jaccard similarity. It divides the size of the intersection by the size of the union, then converts the result into a percentage.

7. Can I use words instead of numbers?

Yes. Elements can be numbers, words, or short labels. Separate items with commas or semicolons, and keep each element written consistently.

8. Why does the calculator warn about universe mismatches?

A complement only makes sense relative to a universe containing the set. If some set elements lie outside the universe, complement-based checks become unreliable.