# Discrete Math Rules—For Your Reference

Below you will find the laws of propositional logic, the rules of inference, and the quantified statements rules of inference presented in a table format.

Law Name | Law | |
---|---|---|

De Morgan’s laws: | ¬( p ∨ q ) ≡ ¬p ∧ ¬q | ¬( p ∧ q ) ≡ ¬p ∨ ¬q |

Idempotent laws: | p ∨ p ≡ p | p ∧ p ≡ p |

Associative laws: | ( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) | ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r ) |

Commutative laws: | p ∨ q ≡ q ∨ p | p ∧ q ≡ q ∧ p |

Distributive laws: | p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) | p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ) |

Identity laws: | p ∨ F ≡ p | p ∧ T ≡ p |

Domination laws: | p ∧ F ≡ F | p ∨ T ≡ T |

Double negation law: | ¬¬p ≡ p | |

Complement laws: | p ∧ ¬p ≡ F ¬T ≡ F | p ∨ ¬p ≡ T ¬F ≡ T |

Absorption laws: | p ∨ (p ∧ q) ≡ p | p ∧ (p ∨ q) ≡ p |

Conditional identities: | p → q ≡ ¬p ∨ q | p ↔ q ≡ ( p → q ) ∧ ( q → p ) |

Rule of inference | Name |
---|---|

p p → q∴ q |
Modus ponens |

¬q p → q∴ ¬p |
Modus tollens |

p ∴ p ∨ q |
Addition |

p ∧ q ∴ p |
Simplification |

p q∴ p ∧ q |
Conjunction |

p → q q → r∴ p → r |
Hypothetical syllogism |

p ∨ q ¬p∴ q |
Disjunctive syllogism |

p ∨ q ¬p ∨ r∴ q ∨ r |
Resolution |

Rule of Inference | Name | Example |
---|---|---|

c is an element (arbitrary or particular ∀x P(x)∴ P(c) |
Universal instantiation | Sam is a student in the class. Every student in the class completed the assignment. Therefore, Sam completed his assignment. |

c is an arbitrary element P(c)∴ ∀x P(x) |
Universal generalization | Let c be an arbitrary integer. c ≤ c ^{2}Therefore, every integer is less than or equal to its square. |

∃x P(x)∴ (c is a particular element) ∧ P(c) |
Existential instantiation | There is an integer that is equal to its square. Therefore, c ^{2} = c, for some integer c. |

c is an element (arbitrary or particular) P(c)∴ ∃x P(x) |
Existential generalization | Sam is a particular student in the class. Sam completed the assignment. Therefore, there is a student in the class who completed the assignment. |

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