# 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:

$a$ | $b$ | $a\vee b$ |
---|---|---|

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 “$(a\wedge b)\to c$” would be

$a$ | $b$ | $c$ | $(a\wedge b)\to c$ |
---|---|---|---|

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 $\wedge $ represents logical and, while $\to $ represents the conditional truth function).

To compute truth tables of expressions, one often proceeds in steps. for instance, to compute a truth table for “$\mathrm{\neg}p\vee (p\wedge q)$, one might proceed as follows:

$p$ | $q$ | $\mathrm{\neg}p$ | $(p\wedge q)$ | $\mathrm{\neg}p\vee (p\wedge q)$ |
---|---|---|---|---|

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:

$p$ | $q$ | $p\vee q$ | $p\wedge q$ | $p\u22bbq$ | $p\to q$ | $p\leftrightarrow q$ |
---|---|---|---|---|---|---|

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:

$p$ | $q$ | $p\overline{)\phantom{\rule{-1.66666666666667pt}{0ex}}}\wedge q$ | $p\overline{)\phantom{\rule{-1.66666666666667pt}{0ex}}}\vee q$ | $p\leftarrow q$ | $p\to \u0338q$ | $p\overline{)\phantom{\rule{-1.66666666666667pt}{0ex}}}\leftarrow q$ |
---|---|---|---|---|---|---|

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 |

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 |