San Jose State University Department of Mathematics

Math 108 Introduction to Proofs

Catalog Description

The purpose of this course is to develop students' mathematical maturity and skill with proofs. Material covered will include logic; set theory including functions, relations, and cardinality; the real number system, including the completeness axiom; and selected topics.

Prerequisites

Math 32 and Math 42 (with a grade of "C-" or better in each), or instructor consent.

Textbook

Instructor option. Both Mathematical Thinking: Problem Solving and Proofs by D’Angelo and West and Introduction to Advanced Mathematics by Barnier and Feldman have been used.

Course Objectives

To learn what constitutes a valid argument and to be able to utilize a variety of proof styles to give valid proofs and disproofs

Student Outcomes

A student should be able to

1. Determine whether an argument is valid or not. 2. Be able to utilize and appreciate the role of definitions in giving proofs. 3. Be able to give direct proofs. 4. Be able to give proofs by contradiction. 5. Be able to give proofs by induction including strong induction. 6. Be able to prove or disprove statements involving quantifiers.

Course Topics The instructor has some latitude in selecting particular topics and the topics may be limited by the text. The course should, however, include a review of truth tables, notion of statements and predicates, quantified statements, set theory including cardinality, and considerable attention to the proof types listed above. Ideally the course will introduce some topics from advanced courses in the department by way of example of proving from the definition, for example. In introducing proof types most texts stick to topics that are familiar to the student. This is a somewhat artificial setting and in fact students can be confused as to what is fair to assume. The introduction to a few advanced topics gets away from this problem enabling the student to truly be in a setting where only a definition is known. Further, introduction of such topics may be a way for a student to decide whether to major in math or not or whether to go for the BA or the BS.

Postscript

Essentially the entire text Introduction to Advanced Mathematics by Barnier and Feldman has been covered. This is probably a bit much. Students liked the text for the most part. There are many practice problems whose solutions are given. Proofs of odd numbered problems are in the back so be warned. The text has the advantage of introducing number theory, abstract algebra, and real analysis. In fact, there are exercises in the set theory chapter related to the definition of a topology. The chapter on cardinality is poorly done in the opinion of this writer. It seems full of minor results. I suggest replacing with handouts. There is some nonstandard terminology, notation, and assumptions. I would choose a different text but was satisfied enough to use for a second semester.

It is critical that many proofs be turned in and graded.

Spring 2006, Hamann

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