COURSE DESCRIPTION

This course was designed as a transition course between introductory and more advanced mathematics which reinforces certain calculus ideas and emphasizes mathematical reasoning and proof. It serves as an introduction to abstract mathematics as well as topics applicable to computer mathematics. Topics include mathematical logic, informal set theory, relations, functions, network theory, and proof strategies.

We shall emphasize the theoretical underpinnings of the subject, without losing sight of the rich collection of concrete examples and applications, which serve to motivate the abstraction and illuminate the theory.

The construction of proofs is an important thread running through all topics in this course. Indeed, most of the problems (exercises) are of the form, "Prove that...." These exercises are an important part of the course, and it is inconceivable that a student could do well without having seriously attempted a significant number of them.

RATIONAL

The purpose of this course is to provide a bridge from the "solve this problem" orientation of the freshman and sophomore years to the "why is this true?" questions of the junior, senior and beyond student experiences. We wish to provide you with the beginning steps of converting you from a student who uses mathematics to a student who understands mathematics. In mathematics the goal is to seek the truth. Finding answers to mathematical questions is important, but we cannot be satisfied only with this. We must be certain that we are right and that our explanation for why we believe we are correct is convincing to others. The reasoning we use to proceed from what we know to what we wish to show must be logical.

PREREQUISITE Mathematics 251

MATERIALS

A. Textbooks    - Introduction to Abstract Mathematics by Jon F. Lucas and Mathematical Proofs by Gary Chartrand

B. Supplemental Readings - reading assignments from standard references, as well as from selected journal articles

C. Notebook - 3-ring binder for class notes and assignments

ACADEMIC INTEGRITY POLICIES

Students are expected to attend all class sessions. Missing 20% or more of such sessions may result in an automatic failing grade. Further information regarding academic or academically related misconduct, and disciplinary procedures and sanctions regarding such misconduct, may be obtained by consulting the NSU Student Handbook.

COURSE OBJECTIVES

1. The student will understand the place of axioms and primitive undefined words in a mathematical system.

2. Students will be expected to read, understand, and be able to construct proofs.

3. Students will be expected to explain the theoretical (logical) basis for various proof techniques including proof by contradiction, proof by cases, induction proofs, existence proofs, and proof of conditional statements.

4. Students will be able to use the pick a point method to prove that two sets are equal.

5. The student will be able to apply the concepts of set theory and discrete mathematics to practical problems.

STUDENT OBJECTIVES

The student, with 80% accuracy, will be able to:
    give examples of sets
    prove the two sets are equal given a set equality
    determine the cardinal number of finite and infinite sets
    given a statement with two or more variables write out the corresponding truth table
    given a tautology prove that the statements are equivalent
    given a set of premises and conclusion construct a valid proof
    logically deduce that an indirect proof is equivalent to a direct proof
    construct valid induction proofs
    construct valid indirect proofs
    determine whether a graph is Eulerian
    determine whether a graph is Hamiltonian
    apply the appropriate algorithms to a traveling salesman problem
    apply Kruskal's algorithm to a minimal connector problem
    find the chromatic number of a graph
    apply graph coloring to a scheduling problem

GRADES
 

 Grading Scale: A 100-90    B 89-80    C 79-70    F 79 -

SCHEDULE OF TOPICS

Week	Topic	Sections
1	Elementary Logic: Sentences and Symbols and Truth Values	Chapter 1 Sec 1 & 2
2	Elementary Logic: Tautology and Equivalence, Conditional Forms, and Quantifiers	Chapter 1 Sec 3, 4 & 5
3	Test 1 & Mathematical Proofs: Inference and Deduction and Conditional Proof	Chapter 2 Sec 1 & 2
4	Mathematical Proofs: Indirect Proof, Proof by Cases, and Existence Proofs	Chapter 2 Sec 3, 4, & 5
5	Mathematical Proofs: Mathematical Induction both parts	Chapter 2 Sec 6 & 7
6	Test 2 & Set Theory: Sets and Operations and Counting	Chapters 3 Sec 1 & 2
7	Set Theory: Cardinality and Classification Problems	Chapter 3 Sec 3 & 4
8	Set Theory: Algebra of Sets and Proofs of Set Theorems	Chapter 3 Sec 5 & 6
9	Relations: Binary and Equivalence Relations	Chapter 4 Sec 1 & 2
10	Test 3 & Relations: Partitions and Order Relations	Chapter 4 Sec 3 & 4
11	Graphs	Chapters 4 Sec 5
12	Graphs and supplementary material from Intro. to Graphs by Chartrand -Books on reserve at Library	Chapters 4 Sec 5
13	Relations: Inequality Proofs and Divisibility Proofs	Chapter 4 Sec 6 & 7
14	Test 4 &  Functions and review	Chapter 5
15	FINAL EXAM