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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:
| $a$ |
$b$ |
$a \lor b$ |
| F |
F |
F |
| F |
T |
T |
| T |
F |
T |
| T |
T |
T |
For $n$ input variables, there will always be $2^n$ rows in the truth table. A sample truth table for ``$(a \land b) \rightarrow c$ '' would be
| $a$ |
$b$ |
$c$ |
$(a \land b) \rightarrow c$ |
| 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 $\land$ represents logical and, while $\rightarrow$ represents the conditional truth function).
To compute truth tables of expressions, one often proceeds in steps. for instance, to compute a truth table for ``$\neg p \lor (p \land q)$ , one might proceed as follows:
| $p$ |
$q$ |
$\neg p$ |
$(p \land q)$ |
$\neg p \lor (p \land q)$ |
| 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:
| $p$ |
$q$ |
$p \lor q$ |
$p \land q$ |
$p \veebar q$ |
$p \rightarrow q$ |
$p \leftrightarrow q$ |
| 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:
| $p$ |
$q$ |
$p \not\!\!\land q$ |
$p \not\!\lor q$ |
$p \leftarrow q$ |
$p \not\rightarrow q$ |
$p \not\!\leftarrow q$ |
|
|
| 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 |
|
|
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