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Hometruth table

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# 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:

$a$ | $b$ | $a\lor b$ |
---|---|---|

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 |

## Mathematics Subject Classification

03-00*no label found*34C29

*no label found*

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