A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a non-zero denominator.

## Definition of Even

An integer n is even if, and only if, n equals twice some integer.

## Rules of Inference

A rule of inference is a form of argument that is valid.

Modus Ponens | |

Modus Tollens | |

Conjunction | |

Contradiction | |

Transitivity | |

Proof by Division into Cases | |

Generalization | |

Specialization | |

Elimination |

## Logical Equivalences

Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences hold.

Commutative laws | ||

Associative laws | ||

Distributive laws | ||

Idempotent laws | ||

Absorption laws | ||

De Morgan’s laws | ||

Identity laws | ||

Universal Bound laws | ||

Negation laws | ||

Double Negative law | ||

Negations of t and c |

## Biconditional

Given statement variables p and q, the biconditional of p and q is “p if, and only if, q.” If both p and q have the same truth values the statement is true, and it is false if p and q have opposite truth values.

T | T | T |

T | F | F |

F | T | F |

F | F | T |