What Is Proof Table?
A proof table is a truth table used to prove something about a statement. By listing every combination of true and false for the variables, it shows whether the statement is always true, always false, or depends on the inputs.
Type your statement using single-letter variables and the operators ¬, ∧, ∨, ⊕, → and ↔, or the keyboard forms !, &, |, ^^, -> and <->. The table shows the variables first, then each sub-statement from the inside out, and finally the whole expression.
Add a second statement to test logical equivalence. Two statements are equivalent when their final columns match in every row, which is how you prove laws such as the contrapositive and De Morgan's laws.
How to Use the Proof Table Calculator
- Type the statement using single-letter variables.
- Optionally type a second statement to test whether the two are equivalent.
- Read the table from the inner parts to the final column.
Proof Table Formula
Implication
Contrapositive
De Morgan
Proof Table Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p, q, r | Propositional variables, each true or false | |
| ¬p | Not p: true when p is false | |
| p ∧ q | And: true only when both are true | |
| p ∨ q | Or: true when at least one is true | |
| p → q | Implies: false only when p is true and q is false |
Proof Table Worked Example
With the example values loaded in the calculator above, the steps are:
- Variables: p, q, r, so the table has 8 rows.
- Columns are built from the inside out: p → q , q → r , (p → q) ∧ (q → r) , p → r , ((p → q) ∧ (q → r)) → (p → r).
- The last column, ((p → q) ∧ (q → r)) → (p → r), is true in 8 of 8 rows, so it is a tautology.
Result: Classification = Tautology (always true); Rows where the expression is true = 8 of 8.
Worked example: hypothetical syllogism
The default statement (p → q) ∧ (q → r) → (p → r) has 3 variables and 8 rows. The final column is true in all 8, so it is a tautology. That is the proof that if p implies q and q implies r, then p implies r.
Testing equivalence with a second statement
Two statements are logically equivalent if they have the same truth value in every row. Enter p → q in the first box and ¬q → ¬p in the second to confirm the contrapositive.
Where Proof Table Is Used
Discrete maths homework
Check a proof table by hand against the full table.
Proving logical laws
Show that De Morgan's laws or the contrapositive hold in every row.
Simplifying conditions
Test whether two if-conditions in code are equivalent.
Common Proof Table Mistakes to Avoid
- Use brackets whenever you mix operators. Without them, ∧ binds tighter than ∨, which binds tighter than →.
- The table grows as 2 to the power of the number of variables.
Proof Table Terms and Definitions
- Tautology
- A statement that is true under every assignment of truth values.
- Contradiction
- A statement that is false under every assignment.
- Contingency
- A statement that is true for some assignments and false for others.
Proof Table FAQs
What is a proof table?
It is a truth table used to prove something about a statement, such as that it is a tautology or that two statements are equivalent.
How many rows does a truth table have?
2 to the power of n, where n is the number of different variables. Three variables give 8 rows.
How do I know a statement is a tautology?
Its final column is true in every row.
When is p → q false?
Only when p is true and q is false.
How do I show two statements are equivalent?
Build the table for both. If their final columns match in every row, they are equivalent.
Which symbols can I type?
Use ¬ ∧ ∨ ⊕ → ↔, or the keyboard forms ! & | ^^ -> <->.