Calculator Form
Example Data Table
| Example | Set | Relation Pairs | Expected Outcome |
|---|---|---|---|
| Divisibility | {1, 2, 3, 6} | (1,1), (2,2), (3,3), (6,6), (1,2), (1,3), (1,6), (2,6), (3,6) | Valid partial order |
| Mutual Pair Conflict | {a, b, c} | (a,a), (b,b), (c,c), (a,b), (b,a) | Not antisymmetric |
| Missing Consequence | {x, y, z} | (x,x), (y,y), (z,z), (x,y), (y,z) | Not transitive because (x,z) is missing |
Formula Used
Partial order test: A relation R on a set A is a partial order when all three conditions below are true.
Reflexive: ∀a ∈ A, (a, a) ∈ R
Antisymmetric: If (a, b) ∈ R and (b, a) ∈ R, then a = b
Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
Decision rule: Partial Order = Reflexive ∧ Antisymmetric ∧ Transitive
Comparability ratio: Comparable unordered pairs ÷ total unordered pairs × 100
Relation density: |R| ÷ |A×A| × 100
How to Use This Calculator
- Enter the elements of your set using commas or separate lines.
- Enter each ordered pair in the relation. Use formats like (a,b), a,b, or a b.
- Choose whether to display closure details and the relation matrix.
- Click Analyze Relation to test the relation immediately.
- Review the result summary above the form for properties, violations, and extremal elements.
- Inspect the matrix and charts to understand relation structure visually.
- Use the CSV or PDF buttons to export the computed results.
FAQs
1. What does this calculator verify?
It checks whether a finite relation is reflexive, antisymmetric, and transitive. When all three hold, the relation qualifies as a partial order.
2. What input format should I use for pairs?
You can enter pairs as (a,b), a,b, or a b. Separate different pairs using new lines or semicolons.
3. What happens if I repeat a pair?
Duplicate pairs are removed automatically. The relation is analyzed using unique ordered pairs only.
4. Why is my relation not reflexive?
A reflexive relation must include every self pair, such as (a,a), for each element in the set. Missing even one self pair breaks reflexivity.
5. Why can antisymmetry fail?
Antisymmetry fails when two different elements relate both ways. If (a,b) and (b,a) both appear for a ≠ b, the condition is violated.
6. What does a transitivity warning mean?
It means the calculator found a chain like aRb and bRc, but the implied pair aRc is missing. That missing consequence violates transitivity.
7. What are minimal and maximal elements?
Minimal elements have nothing strictly below them. Maximal elements have nothing strictly above them. A set may contain several of each.
8. Can this tool repair a relation automatically?
It suggests reflexive and transitive additions. However, if cycles create antisymmetry conflicts, additions alone cannot turn the relation into a partial order.