Math 305 - Introduction to Advanced Mathematics Fall 2010
Instructor: Elizabeth Meckes
Office: Yost 208
Phone: 368-5015
Email: ese3 [at] cwru.edu
Office Hours: TBA
Textbook:Proof: Introduction to Higher Mathematics, 5e
by W. Esty and N. Esty. The book is only available through the University
Bookstore.
Topics:
We will cover all of chapters 1 through 5 of the text,
with additional
material from chapters 6 through 8 if time permits.
The topics of chapters 1 through 5 are:
Introduction to proofs: sets, logic,
equivalences, negations, tautologies
Sentences with variables: existence statements and negations,
theorems and definitions
Proofs: Inequalities, absolute value, theory of proofs, induction,
proof by contradiction
Set theory: basic set theory, bounds (infima, suprema, etc.)
Functions: definitions, properties, cardinality
Chapers 5 through 8 are introductions to number theory, group theory, and
topology. We will cover some of this material mainly as further practice in
reading and writing about mathematics.
Course Goals: By the end of the term, you should have learned to:
Read critically and with comprehension from an upper-level mathematics
text book. This means being able to explain what you read to a classmate
and answer his/her questions.
Use the basic language of mathematics, so that you can clearly
and correctly articulate your own ideas, both verbally and in writing.
Understand the structure of mathematics: the importance of
definitions, axioms, theorems, proofs, and their relationships.
Feel comfortable with standard patterns of thought used
in mathmematics, especially different methods of proof.
Produce a logically reasoned argument.
Understand the basic concepts of set theory and the theory of
functions.
Class format:
The first half to two-thirds of each class will be in lecture format.
You are expected to attend and take notes! Taking notes and
listening actively to the lecture at the same time is an important skill
to learn for success in future math classes. For many people, taking notes
helps keep them engaged in the lecture, and being able to look at one's own
notes later is a helpful additional association for retaining the material.
Furthermore, some advanced courses do not have a text book, so you will need
to take notes in order to have a reference for the course -- now is the time
to learn to do that well!
After the lecture, the remainder of each class will be devoted to
group work. You will be divided into groups of roughly three students
at the beginning of the term. The text book has problems specifically
intended for group work during class. During these periods, you will
be given a problem to work on with your group. Once you have solved
it, you will write out a careful solution to be turned in. If you
can't solve it, you will write out a detailed explanation of what you
tried and why it didn't work. One group will be chosen at random to
present their work to the class.
Grading: There will be one midterm (worth 25% of the course grade),
one final (worth 40%), weekly homework assignments (worth 25%),and the group assignments (10%).
Homework Problems:
Homework is due at the beginning of class; assignments are
posted below.
The homework is probably the single biggest factor in determining
how much you get out of the course, so if you're having trouble, please come
ask questions. Correctness and clarity of exposition
are a major focus here and count
just as much as correctness of content. You may discuss the homework with
other students, however, you must write up solutions on your own.
Assignments:
For 9/27: Section 1.6, problems A2, A17, A52, B8, B25, B41, B43; Section
2.1 problems A16(a,c), B20, READ problems C23-C24 (you don't have to turn
anything in); Section 2.2, problems A3, A17, class problem.
For 10/4: Section 2.2, problems B4, B19, B25, B39, B62(a,b), B87, C11.
Section 2.3, problems A14, A19, A32(a,d), B22, B60, B87, C3.
Section 2.5, problems B23, B37, C2 (you may assume that there exists
at least one irrational number).
For 10/11: Section 3.1, problems A7, A9, B4, B16, B40 (in some students' books, this seems to be B43; the one about the square root of x+y), C1. Section 3.2,
problems A29, B27 (not by cases). Section 3.3, problems A4, A19, A22,
B3, B36, B47.