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# Exam 1

### MAT 300 Mathematical Structures

Question | Answer |
---|---|

Def: Tautology | Formulas that are always true ex) P or not P |

Def: Contradiction | Formulas that are always false ex) P and not P |

Def: Converse | The reversal of an if-then statement. Logically equivalent to inverse. ex) P -> Q ; Q -> P |

R | The set of all real numbers, containing all rational and irrational numbers. Q, N , Z are the subsets of the set R. |

Q | The set of all rational numbers. The rational numbers are any number which could be written like p/q, where p and q are integers. |

N | The set of all natural numbers like:1,2,3,4,5,....... The set has starting number 1 and the consecutive numbers increments by 1 . It has no end. |

Z | Integers (positive or negatives including zero). |

Def: Statement | An assertion (about mathematics) that is objectively either true or false |

Def: Argument | A sequence of statements. All except the last are premises, while the last is a conclusion. |

Def: Valid Argument | P1, P2, P3, ... , Pk | Q Q is true when all Ps are true (conclusion is true whenever all premises are true |

Def: Proof of an Assertion Q | A sequence of true statements connected with valid arguments and ending with Q |

Def: Equivalence | Two statements with the same truth table Symbol: <=> |

Def: Set | A collection of objects (elements) |

Free Variables | Truth value depends on variable |

Bound Variables | Variable can be replaced or eliminated |

DeMorgan's Law | ~(A or B) --> ~A and ~B ~(A and B) --> ~A or ~B |

Def: Contrapositive | Reversing if-then and adding "not" to both sides (combines inverse and converse). Logically equivalent to conditional! ex) P --> Q ; ~Q --> ~P |

Def: Inverse | Adds not to both sides. Logically equivalent to converse. |

Def: Power Set | The set whose elements are all subsets of A P(A) = { x | x is a subset of A } |

Commutative Laws | P ^ Q is equivalent to Q ^ P P or Q is equivalent to Q or P |

Associative Laws | When a statement has all "and"s or "or"s, it doesn't matter how you separate them P ^ (Q ^ R) is equivalent to (P ^ Q) ^ R P or (Q or R) is equivalent to (P or Q) or R |

Idempotent Laws | P and/or P is equivalent to P |

Distributive Laws | P ^ (Q or R) is equivalent to (P ^ Q) or (P ^ R) P or (Q ^ R) is equivalent to (P or Q) ^ (P or R) |

Absorption Laws | P or (P ^ Q) is equivalent to P P ^ (P or Q) is equivalent to P |

Modus Ponens | If you know that P is true and P --> Q is true, then Q is also true |

Modus Tollens | If you know that P --> Q is true and Q is false, then Q is false |

Proof by Contradiction | Show that if a proposition is false, a contradiction is implied |

Def: Elements | Objects in sets |