Calculator Inputs
Formula Used
Single-variable universal quantifier
For a finite domain D, the statement ∀x ∈ D, P(x) is true exactly when every tested value makes P(x) true.
Equivalent finite form: ∀x ∈ D, P(x) ⇔ ∧x∈D P(x)
Two-variable universal quantifier
For finite domains D₁ and D₂, the statement ∀x ∈ D₁ ∀y ∈ D₂, R(x,y) is true when every ordered pair satisfies R.
Equivalent finite form: ∀x∀y R(x,y) ⇔ ∧x∈D₁ ∧y∈D₂ R(x,y)
Counterexample test
A universal statement is false if at least one counterexample exists.
Truth ratio: (number of true cases ÷ total tested cases) × 100
How to Use This Calculator
- Choose unary mode for ∀xP(x) or binary mode for ∀x∀yR(x,y).
- Enter a finite domain using values or range notation.
- Select the predicate or relation you want to test.
- Fill in constants k1 and k2 if your choice needs thresholds, bounds, or remainders.
- Submit the form to see the conclusion, counterexamples, graph, and truth table.
- Use CSV or PDF export to save the tested statement and results.
Example Data Table
| Example | Statement | Domain | Result | Counterexample |
|---|---|---|---|---|
| 1 | ∀x ∈ {2,4,6,8}, x is even | 2, 4, 6, 8 | True | None |
| 2 | ∀x ∈ {1,2,3,4}, x > 0 | 1, 2, 3, 4 | True | None |
| 3 | ∀x ∈ {1,2,3,4}, x is prime | 1, 2, 3, 4 | False | 1, 4 |
| 4 | ∀x∀y with D={1,2,3}, x ≤ y | (1,2,3) × (1,2,3) | False | (2,1), (3,1), (3,2) |
FAQs
1) What does a universal quantifier mean?
It means “for every value in the domain.” The statement is true only when each tested element, or each ordered pair in binary mode, satisfies the chosen predicate or relation.
2) Why can an empty domain still return true?
In classical logic, a universal statement over an empty domain is vacuously true. There is no element available to violate the predicate, so no counterexample exists.
3) What is a counterexample here?
A counterexample is any tested value, or any ordered pair, that makes the predicate false. Finding even one counterexample is enough to make the universal statement false.
4) Can I test pairwise relations like x ≤ y?
Yes. Choose binary mode and select a relation such as x ≤ y, x + y = k1, or |x − y| ≤ k1. The calculator then checks all ordered pairs.
5) Why do prime or parity tests fail for decimals?
Prime, even, odd, and composite properties are defined for integers in this calculator. Decimal inputs do not count as valid prime or parity candidates, so those tests return false.
6) What does the truth ratio show?
It shows the percentage of tested cases that satisfy the predicate. A ratio below 100% means the universal statement failed somewhere in the finite domain.
7) Can this calculator prove statements on infinite sets?
No. It evaluates finite domains that you provide. That makes it useful for examples, classroom practice, counterexample hunting, and checking finite cases in structured proofs.
8) Why does the graph help?
The graph quickly highlights where truth coverage breaks. In unary mode you see failing elements. In binary mode you see failing cells across the relation matrix.