Schedule

Induction proofs for divisibility and order.

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

Th. 1/20 | Introduction. Statements, negation, conjunction,
disjunction. Logical equivalences |
Sec. 1.1 | |

Tu. 1/25 | Logical equivalences, tautologies, contradictions.
Conditionals and biconditionals. |
Secs. 1.1-2 | |

Th. 1/27 | Conditionals and English usage.
Logical arguments. |
Secs. 1.2-3 | |

Tu. 2/1 | Logic circuits and logic puzzles. Sets |
Secs. 1.4, 5.1 | |

Th. 2/3 | Sets, complement, intersection, union. Predicates |
Secs. 5.1, 2.1-2 | |

Tu. 2/8 | Predicates and English.Truth set of a predicate.
Quantified predicates. |
Secs. 2.1-3 | |

Th. 2/10 | Multiply quantified predicates. Arguments with quantified predicates. |
Secs. 2.3 (2nd Ed) 2.4 (3rd Ed.) | |

Tu. 2/15 | Sets: Cartesian products, power set, partitions. Proofs: element-wise proofs, algebraic proofs. |
Secs. 5.1-3 | |

Th. 2/17 | More proofs of set theorems.
Mathematics: Axiom, Definition, Theorem, Proof Axioms for the integers. |
Secs. 5.2-3 3.1 | |

Tu. 2/22 | Fundamental properties of the integers.
Order, divisibility. |
Secs. 3.1, 3.3, 3.4 | |

Th. 2/24 | EXAM | Ch. 1, 2, 5 | |

Tu. 3/1 | The quotient-remainder theorem. The Euclidean algorithm. |
Secs. 3.1, 3.3, 3.5 | |

Th. 3/3 | Representation of integers in binary, octal etc. | Secs. 1.4, 4.1 (just one page) | |

Tu. 3/8 | Primes and Unique Factorization. Infinitude of primes. | Sec. 3.5-7 | |

Th. 3/10 | Rational numbers and real numbers. The floor and ceiling functions. |
Secs. 3.5-7 | |

Tu. 3/15 | Sequences, summation and product notation. The well ordering principle. The principle of induction. Proof by induction. |
Sec. 4.1-2, 4.4 | |

Th. 3/17 | Recursively defined functions. | Sec. 4.2-3, 8.1 | |

Tu. 3/22 | Sec. 4.2-3, 8.1 | ||

Th. 3/24 | Strong induction. | Sec. 4.4, 8.1-2 | |

Tu. 4/5 | Questions before the exam? Functions |
Sec. 7.1-2. | |

Th. 4/7 | EXAM. | Secs. 3.1-7, 4.1-4; 8.1-2. | |

Tu. 4/12 | Relations Functions: one-to-one, onto. |
Sec. 10.1 (except last two pages) Sec. 7.1-2. | |

Th. 4/14 | Functions: composition. | Sec. 7.4 | |

Tu. 4/19 | Relations on a set: reflexive, symmetric, transitive
Partitions and equivalence relations. |
Sec. 10.1 (last two pages), 10.2 | |

Th. 4/21 | Examples of equivalence relations.
Partially ordered sets. |
Sec 10.3 | |

Tu. 4/26 | Counting: addition, multiplication rules, inclusion-exclusion. |
Sec 6.2,3 | |

Th. 4/28 | Four ways to count. | Sec 6.4. | |

Tu. 5/3 | From counting to probability and gambling. | Sec. 6.1,4 | |

Th. 5/5 | EXAM. | Secs 7.1,2,4 (1,3,5 in 2nd Ed.) Secs 10.1,2,3,5. | |

Tu. 5/10 | Binomial theorem.
Pascal's triangle. Questions? |
Sec. 6.6, 6.7 | |

Tu. 5/17 (8:00-10:00) | FINAL EXAM. | Cumulative. |