Calculator Input
Enter two logical expressions. Supported operators include NOT, AND, OR, XOR, IMPLIES, IFF, parentheses, and symbols like !, &, |, ->, and <->.
Example Data Table
| Expression A | Expression B | Expected Outcome | Reason |
|---|---|---|---|
| !(P AND Q) | (!P OR !Q) | Equivalent | De Morgan's law gives identical truth values. |
| (P -> Q) | (!P OR Q) | Equivalent | Implication expands into a disjunction form. |
| (P XOR Q) | ((P OR Q) AND !(P AND Q)) | Equivalent | XOR is true only when inputs differ. |
| (P -> Q) | (Q -> P) | Not Equivalent | Direction matters in implication statements. |
Formula Used
Logical equivalence means two expressions return the same truth value for every possible variable assignment.
Main rule: A ≡ B if and only if Val(A, v) = Val(B, v) for every valuation v.
- Negation: NOT P is true when P is false.
- Conjunction: P AND Q is true only when both are true.
- Disjunction: P OR Q is true when at least one is true.
- Exclusive OR: P XOR Q is true when exactly one is true.
- Implication: P → Q is equivalent to NOT P OR Q.
- Biconditional: P ↔ Q is true when both values match.
The calculator converts each expression into postfix form, evaluates all truth-table rows, and compares final output columns row by row.
How to Use This Calculator
- Enter the first logical expression in the Expression A box.
- Enter the second logical expression in the Expression B box.
- Choose alphabetical order or appearance order for variables.
- Select your preferred truth output style.
- Choose whether to show all rows or mismatched rows only.
- Press the calculation button to build the truth table.
- Review the equivalence status, summary metrics, chart, and table.
- Export the displayed truth table as CSV or PDF if needed.
Frequently Asked Questions
1. What does logical equivalence mean?
Logical equivalence means two statements always produce the same truth value. If every truth-table row matches, the expressions are equivalent, even when they look different.
2. Why does the calculator build a truth table?
A truth table checks every possible combination of variable values. This guarantees the comparison is complete, not based on only one example or a partial test.
3. Which operators can I use here?
You can use NOT, AND, OR, XOR, IMPLIES, IFF, parentheses, and common symbolic versions such as !, &, |, ^, ->, and <->.
4. Can I compare implication and disjunction forms?
Yes. A common example is P -> Q and !P OR Q. The calculator verifies whether they match in every row and confirms equivalence.
5. Why does variable count matter?
Each added variable doubles the number of truth-table rows. More variables create larger tables, so the page limits variables to keep results readable and practical.
6. What if the result shows mismatched rows?
Mismatched rows mean the expressions are not logically equivalent. Those rows identify the exact variable assignments where the outputs differ.
7. Does expression order affect the answer?
No. Swapping Expression A and Expression B does not change equivalence. The calculator compares both output columns across the same set of assignments.
8. What do the CSV and PDF exports contain?
The exports contain the currently displayed truth-table rows. If you choose mismatched rows only, the downloaded file will include only those rows.