Schedule

Solutions to last HW.

Permutations and combinations.

Pascal's triangle.

Day | Topics | Preparation | |
---|---|---|---|

Th. 1/19 | Introduction. Statements, negation, conjunction,
disjunction. Logical equivalences |
Sec. 2.1 | |

Tu. 1/24 | Logical equivalences, tautologies, contradictions.
Conditionals and biconditionals. |
Secs. 2.1-2 | |

Th. 1/26 | Conditionals and English usage.
Logical arguments. |
Secs. 2.2-3 | |

Tu. 1/31 | Logic circuits and logic puzzles. Sets |
Secs. 2.4, 1.1-2, 6.1-2 (pp.336-344) | |

Th. 2/2 | Sets, complement, intersection, union. Main properties of sets. Predicates and truth sets |
Secs. 6.1-2, 3.1-2 | |

Tu. 2/7 | Predicates and English.
Quantified predicates. Negation. Multiply quantified predicates. |
Secs. 3.1-3 | |

Th. 2/9 | Arguments with quantified predicates: Puzzles Proofs of set properties: element-wise. |
Sec. 3.4 | |

Tu. 2/14 | Algebraic proofs of set theorems.
Disproof by counterexample. Sets: Cartesian products and the multiplication rule. |
Secs. 6.1,2 | |

Th. 2/16 | Sets and counting: Partitions and inclusion/exclusion. Power sets | Secs. 6.1-3 Sec. 6.1 (p. 344-348) Sec. 9.2 (525-529), Sec. 9.3 | |

Tu. 2/21 | Mathematics: Axiom, Definition, Theorem, Proof
Axioms for the integers. Fundamental properties of the integers. Order. |
Sec. 3.1 | |

Th. 2/23 | Integers: Divisibility, linear combinations. The quotient-remainder theorem. |
Secs. 3.1, 3.3, 3.4 | |

Tu. 2/28 | TEST: Logic, sets, counting | Ch. 2, 3, 6 | |

Th. 3/1 | Applications of the quotient-remainder theorem. The Euclidean algorithm. Binary expansion of a number. |
Secs. 4.1, 4.3, Sec. 4.8 | |

Tu. 3/6 | Arithmetic in base .b Primes and the Unique Factorization Theorem. |
3rd E.: Sec. 1.5 (pp. 57-60, 70-73) 4.1 (pp. 211-213) 3.3
(pp. 153-4).
4th Ed.:Sec. 2.5 (pp. 78-81, 91-93), 5.1 (pp. 240-242), 4.3 (pp.176-7) | |

Th. 3/8 | Infinitude of primes. Rational Numbers and real numbers. The floor and ceiling functions. |
Sec. 4.5-7 | |

Tu. 3/13 | Sequences, summation and product notation. The well ordering principle. |
Secs. 5.1-2, 5.4 | |

Th. 3/15 | The principle of induction. Proof by induction. Recursively defined functions. |
Sec. 5.2,6 | |

Tu. 3/20 | Induction proofs for divisibility and order. | Sec. 5.3 | |

Th. 3/22 | Strong Induction. | Sec. 5.4, 6, 7 | |

Tu. 4/3 | Functions and relations. | Sec. 7.1-2. Sec. 8.1 (except last two pages) | |

Th. 4/5 | TEST: Properties of the integers, sequences, induction. | Chs. 4, 7 | |

Tu. 4/10 | Relations and their inverses. Functions: injective (one-to-one), surjective (onto). |
Sec. 8.1 Sec. 7.2 | |

Th. 4/12 | Functions: composition. Relations on a set: reflexive, symmetric, transitive. |
Sec. 7.3 Sec. 8.1 (p. 446), Sec. 8.2 | |

Tu. 4/17 | Equivalence relations and partitions. Examples of equivalence relations. |
Sec 8.3 | |

Th. 4/19 | Partially ordered sets. | Sec 8.5 | |

Tu. 4/24 | Partially ordered sets. The pigeonhole principle. Four ways to count. |
Sec. 8.5 Sec. 9.4. Sec. 9.5 | |

Th. 4/26 | Sec. 9.6,7 | ||

Tu. 5/1 | TEST: Functions, relations, equivalence relations, posets. | Secs 7.1,2,4. Secs. 8.1,2,3,5. | |

Th. 5/3 | Binomial Theorem From counting to probability and gambling. Poker hands. |
Sec. 9.1,5 | |

Tu. 5/8 | Variety show. Questions? |
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Tu. 5/10 (8:00-10:00) | FINAL EXAM. | Cumulative. |