truth table
A truth table
is a tabular listing of all possible input value combinations
for a logical function and their corresponding output values. Similarly, the truth table of a logical proposition
is the truth table of the corresponding logical function.
For instance, the truth table of the connective “or” is as follows:
| a | b | a∨b |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
For n input variables, there will always be 2n rows in the truth table. A sample truth table for “(a∧b)→c” would be
| a | b | c | (a∧b)→c |
|---|---|---|---|
| F | F | F | T |
| F | F | T | F |
| F | T | F | T |
| F | T | T | F |
| T | F | F | T |
| T | F | T | F |
| T | T | F | T |
| T | T | T | T |
(Note that ∧ represents logical and, while → represents the conditional truth function).
To compute truth tables of expressions, one often proceeds in steps. for instance, to compute a truth table for “¬p∨(p∧q), one might proceed as follows:
| p | q | ¬p | (p∧q) | ¬p∨(p∧q) |
|---|---|---|---|---|
| F | F | T | F | T |
| F | T | T | F | T |
| T | F | F | F | F |
| T | T | F | T | T |
For reference, here is a truth table of some popular connectives:
| p | q | p∨q | p∧q | p⊻ | ||
|---|---|---|---|---|---|---|
| F | F | F | F | F | T | T |
| F | T | T | F | T | T | F |
| T | F | T | F | T | F | F |
| T | T | T | T | F | T | T |
For completeness, here are the remaining connectives, excluding trivial connectives which depend on only one or none of their arguments:
| F | F | T | T | T | F | F |
| F | T | T | F | F | F | T |
| T | F | T | F | T | T | F |
| T | T | F | F | T | F | F |
| Title | truth table |
|---|---|
| Canonical name | TruthTable |
| Date of creation | 2013-03-22 11:54:35 |
| Last modified on | 2013-03-22 11:54:35 |
| Owner | rspuzio (6075) |
| Last modified by | rspuzio (6075) |
| Numerical id | 16 |
| Author | rspuzio (6075) |
| Entry type | Definition |
| Classification | msc 03-00 |
| Classification | msc 34C29 |
| Related topic | ZerothOrderLogic |
| Related topic | PropositionalCalculus |