#### Prerequisites & Bulletin Description

#### Course Objectives

The principle aim of Exploration and Proof is to prepare students to understand the role of definitions and theorems in mathematics and to write proofs. They learn to explore concepts as a means for understanding new definitions and theorems that they encounter and for isolating key ideas they need in writing a proof. Also, students learn to use quantifiers to express mathematical ideas precisely, as well as learning to understand mathematical prose that is stated symbolically. Students learn key ideas in basic set theory concerning set operations, the empty set, power sets, subsets, and Cartesian products. They learn key notions in number theory such as prime numbers, multiples, the well-ordering principle, and the division algorithm. Students who successfully complete the course should be able to prove basic theorems concerning injective, surjective functions, and bounded sets. They also become familiar with direct and indirect proof and proof by induction.

#### Evaluation of Students

Students will be graded on their ability to write clear proofs in correct mathematical English, to work with new definitions, and to understand the key concepts covered in class. Their grade will be based on weekly graded homework assignments, in-class or take home examinations and a final examination.

#### Course Outline

Topics |
Number of Weeks |

Introduction to exploration and proofs | 2 |

Exploration and proof with sets | 2 |

Exploration and proof with integers | 2 |

Using quantifiers to express mathematical ideas | 2 |

Bounded sets | 1 |

Functions | 1 |

Proof techniques for different quantifiers | 1 |

Proof by contradicition and contrapositives | 1 |

Principle of mathematical induction | 1 |

Strong Induction and Well-ordering Principle | 1 |

#### Textbooks & Software

Dan Fendel and Diane Resek, *Foundations of Higher Mathematics*, Addison Wesley

Eccles, *An Introduction to Mathematical Reasoning*, Cambridge U. Pr.

Submitted by: Diane Resek

Date: Apr. 28, 2003