Calculator Inputs
Use commas between elements. Supported symbols in expressions: | for union, & for intersection, - or \ for difference, ^ for symmetric difference, ~A or A' for complement.
Formula Used
Union: A ∪ B contains every element appearing in A, B, or both.
Intersection: A ∩ B contains only common elements.
Difference: A − B keeps elements in A that are not in B.
Complement: A′ contains every universal-set element not in A.
Identity Testing: Left side equals right side when both evaluated results match exactly.
Proof Support: The calculator verifies identities computationally for the entered data and presents intermediate operations to explain why a statement holds or fails.
How to Use This Calculator
- Enter the universal set first.
- Fill in sets A, B, C, and D as needed.
- Select a mode: identity check, expression evaluation, or relation check.
- For identity mode, enter two expressions to compare.
- For expression mode, enter one expression to evaluate.
- For relation mode, choose named sets and the relation type.
- Click Solve Now to show the result above the form.
- Review the step summary and chart.
- Use the CSV or PDF button to export results.
Example Data Table
| Example | Universal Set | A | B | C | Test | Outcome |
|---|---|---|---|---|---|---|
| De Morgan Check | {1,2,3,4,5,6,7,8,9,10} | {1,2,3,4,5} | {3,4,5,6} | {2,4,6,8} | ~(A|B) = ~A & ~B | Holds for these sets |
| Expression | {1,2,3,4,5,6,7,8,9,10} | {1,2,3,4,5} | {3,4,5,6} | {2,4,6,8} | (A|B)&C | {2,4,6} |
| Relation | {1,2,3,4,5,6,7,8,9,10} | {1,2,3,4,5} | {3,4,5,6} | {2,4,6,8} | A ⊆ U | True |
Frequently Asked Questions
1. What does this set theory proof solver calculator do?
It evaluates set expressions, checks set relations, and compares two expressions under one universal set. It helps verify identities using actual entered elements and shows computational steps.
2. Does this calculator produce formal symbolic proofs?
It provides computational verification rather than a complete textbook-style symbolic derivation. That still helps students test identities, catch mistakes, and understand intermediate set operations.
3. Which operators can I use in expressions?
Use | for union, & for intersection, - or \ for difference, ^ for symmetric difference, and ~A or A' for complement. Parentheses are supported.
4. Why must every set stay inside the universal set?
Complements are defined relative to the universal set. If a set contains outside elements, complement and identity checks become inconsistent, so the calculator validates that first.
5. How is identity checking performed?
The calculator evaluates the left expression and right expression separately. If both resulting sets match exactly, the identity is treated as verified for the entered dataset.
6. Can I use letters instead of numbers as elements?
Yes. Elements are read as text values, so numbers, words, or symbols can be used. Keep them comma-separated and consistent across all sets.
7. What is the benefit of the chart?
The chart gives a quick size comparison between sets and results. It helps users notice overlaps, reductions, and identity differences faster than reading only tables.
8. When should I export CSV or PDF?
Export CSV when you need structured results for spreadsheets. Export PDF when you want a shareable summary for homework, notes, or documentation.