# Truth table proofs

Truth tables are used to show all possible values that a given logical expression might take. For example, the following gives the definition of the logical AND operation.

A | B | A ∧ B |
---|---|---|

false | false | false |

false | true | false |

true | false | false |

true | true | true |

A truth table can also be used to prove a logical identity. The following proves DeMorgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same.

A | B | A ∧ B | ¬ (A ∧ B) | (¬ A) ∨ (¬ B) |
---|---|---|---|---|

false | false | false | true | true |

false | true | false | true | true |

true | false | false | true | true |

true | true | true | false | false |