Truth Table
Definition
A truth table is a systematic listing of all possible truth values for a logical expression, showing the output for every combination of inputs.
Construction
For \(n\) propositional variables, a truth table has \(2^n\) rows.
Example: AND
| \(P\) | \(Q\) | \(P \land Q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Example: Implication
| \(P\) | \(Q\) | \(P \implies Q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note: \(P \implies Q\) is false only when \(P\) is true and \(Q\) is false.
Related
- Proposition — The basic building block
- Logical Connective — AND, OR, NOT, IMPLIES
- Direct Proof — Using truth tables to prove validity