# Difference between revisions of "Truth table proofs"

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− | Truth tables are used to show all possible values that a given logical expression might take. | + | Truth tables are used to show all possible values that a given logical expression might take. |

+ | |||

+ | =Examples= | ||

+ | ==Definition of AND== | ||

+ | The following gives the definition of the logical AND operation. | ||

{| class="wikitable" | {| class="wikitable" | ||

Line 14: | Line 18: | ||

|} | |} | ||

− | A truth table can also be used to prove a logical identity. The following proves | + | ==Proof of one of De Morgan's laws== |

+ | A truth table can also be used to prove a logical identity. The following proves De Morgan'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. | ||

{| class="wikitable" | {| class="wikitable" | ||

|- | |- | ||

− | ! A !! B !! A ∧ B !! ¬ (A ∧ B) !! (¬ A) ∨ (¬ B) | + | ! A !! B !! A ∧ B !! ¬ (A ∧ B) !! (¬ A) ∨ (¬ B) |

|- | |- | ||

| false || false || false || true || true | | false || false || false || true || true | ||

Line 28: | Line 33: | ||

| true || true || true || false || false | | true || true || true || false || false | ||

|} | |} | ||

+ | |||

+ | ==Disproof of a logical fallacy== | ||

+ | Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal. | ||

+ | |||

+ | Consider this truth table. | ||

+ | {| class="wikitable" | ||

+ | |- | ||

+ | ! A !! B !! A → B (which is equivalent to ¬ A ∨ B) | ||

+ | |- | ||

+ | | false || false || true | ||

+ | |- | ||

+ | | false || true || true | ||

+ | |- | ||

+ | | true || false || false | ||

+ | |- | ||

+ | | true || true || true | ||

+ | |} | ||

+ | In the | ||

+ | |||

+ | |||

+ | =Assignment= |

## Revision as of 15:03, 29 August 2022

Truth tables are used to show all possible values that a given logical expression might take.

## Contents

# Examples

## Definition of AND

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 |

## Proof of one of De Morgan's laws

A truth table can also be used to prove a logical identity. The following proves De Morgan'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 |

## Disproof of a logical fallacy

Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal.

Consider this truth table.

A | B | A → B (which is equivalent to ¬ A ∨ B) |
---|---|---|

false | false | true |

false | true | true |

true | false | false |

true | true | true |

In the