Calculator Inputs
Enter elements separated by commas, semicolons, pipes, tabs, or new lines. Duplicate values are removed automatically because sets keep unique elements only.
Example Data Table
| Example Input | Value | Example Output | Cardinality |
|---|---|---|---|
| U | {1, 2, 3, 4, 5, 6, 7, 8} | n(U) | 8 |
| A | {1, 2, 3, 4} | n(A) | 4 |
| B | {3, 4, 5, 6} | n(B) | 4 |
| C | {4, 6, 7} | n(C) | 3 |
| A ∪ B | {1, 2, 3, 4, 5, 6} | n(A ∪ B) | 6 |
| A ∩ B | {3, 4} | n(A ∩ B) | 2 |
| A △ B | {1, 2, 5, 6} | n(A △ B) | 4 |
| A × B | All ordered pairs | n(A × B) | 16 |
| A ∪ B ∪ C | {1, 2, 3, 4, 5, 6, 7} | n(A ∪ B ∪ C) | 7 |
| A′ in U | {5, 6, 7, 8} | n(A′) | 4 |
Formula Used
Basic Cardinality
The cardinality of a set is the number of distinct elements inside it.
- n(A) = number of unique elements in set A
- n(B) = number of unique elements in set B
- n(C) = number of unique elements in set C
Two-Set Operations
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- n(A ∩ B) = common elements in both sets
- n(A − B) = elements in A but not in B
- n(B − A) = elements in B but not in A
- n(A △ B) = n(A − B) + n(B − A)
Three-Set Inclusion-Exclusion
When set C is added, the calculator uses the standard inclusion-exclusion rule.
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)
Complement, Power Set, and Product
- n(A′) = n(U − A), measured inside universal set U
- |P(A)| = 2n(A)
- n(A × B) = n(A) × n(B)
- Jaccard(A,B) = n(A ∩ B) / n(A ∪ B)
How to Use This Calculator
- Enter elements for set A and set B.
- Add set C if you need three-set analysis.
- Add universal set U if you want complements.
- Choose case-sensitive or case-insensitive comparison.
- Press Calculate Cardinality to generate results.
- Review the summary tiles, normalized sets, graph, and metric table.
- Use the export buttons to save results as CSV or PDF.
FAQs
1) What does set cardinality mean?
Set cardinality means the count of distinct elements in a set. Repeated values do not increase the count because a set stores each unique element only once.
2) Does the calculator remove duplicate entries?
Yes. The calculator automatically removes duplicates from every entered set. It then computes all operations using unique values only, which matches proper set theory rules.
3) Can I use words instead of numbers?
Yes. You can enter numbers, words, codes, or mixed labels. The calculator treats each item as a set element and compares them exactly or without case, based on your selected mode.
4) When should I enter a universal set?
Enter a universal set when you need complements such as A′, B′, or C′. Complements are always measured relative to the universal set you provide.
5) What is the symmetric difference?
The symmetric difference contains elements that appear in exactly one of the two sets. It excludes everything found in their intersection.
6) What does the power set count show?
The power set count shows how many subsets a set has. For a set with n unique elements, the total number of subsets equals 2 raised to n.
7) Why is the Cartesian product cardinality so large?
The Cartesian product pairs every element of one set with every element of another. Its size grows as n(A) multiplied by n(B), so larger inputs increase it quickly.
8) Can I export the computed results?
Yes. After calculation, you can download a CSV file for spreadsheet use or a PDF report for sharing, printing, or storing your results.