## Mathematical Method, The: A Transition to Advanced Mathematics, 1/e

Murray Eisenberg, University of Massachusetts, Amherst

Published August, 1995 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-127002-8

mailings
on this subject.

A “gentle” and superbly written introduction to mathematical rigor and proof, this text is designed to prepare students (especially math majors) for subsequent junior/senior-level courses in Abstract Analysis and Algebra by providing extended developments of several basic mathematical topics. It offers a finely-focused, “hands-on” approach to the rigors of mathematical reasoning — “the mathematical method” — i.e., developing carefully formulated definitions, clearly stated assumptions, and logically rigorous proofs.

introduces fundamentals (e.g., logic, sets, relations, functions, etc.) in context with “real” mathematical topics — induction and recursion, number theory, axiomatic development of real numbers, and cardinality.

• Rather than giving isolated extracts of mathematics to illustrate a catalog of proof methods, it uses extended developments of those topics in which various techniques of proof are employed — e.g., equivalence relations and equivalence classes are introduced through congruence of integers and the algebra of congruence classes.
begins topics with fairly simple ideas and develops them sufficiently to reach non-trivial and/or substantial results — e.g. uniqueness is developed up to isomorphism of the field of real numbers.
offers an alternative approach to topics which overlap those in subsequent abstract analysis and algebra courses — e.g., the axiomatic treatment of the reals as an ordered field that is Archimedean, and has the nested interval property (instead of being order complete in terms of least upper bounds).
treats recursion carefully and correctly — to avoid the mystification that typically results from casual treatment of recursive definition.
makes explicit what is being assumed about natural numbers and integers (Appendix D).
uses the notation and terminology of functions (explained in Appendix B) where appropriate throughout — to help students obtain experience with the critical notion of function in a variety of contexts.
intersperses problems through the narrative.
• exercises range from very easy, immediate applications of the definitions, through routine and moderately challenging problems, to multi-part, substantial problems suitable for “mini-projects.”
• features several optional examples and exercises that involve writing a computer program or defining a function in a computer algebra system such as MATHEMATICA.

1. Induction.

Ordinary Induction. Recursion. Summation. Well–ordering and Strong Induction. Binomial Coefficients.

2. Number Theory.

The Division Theorem. Divisibility. Prime Numbers. Congruence. Congruence Classes.

3. The Real Numbers.

Fields. Ordered Fields. Nested Intervals and Completeness. Isomorphism of Fields. Null Sequences and Limits. The Complex Number Field.

4. Cardinality.

Finite and Infinite Sets. Countable and Uncountable Sets. Uncountable Sets.

Appendix A. Sets.

Elements and Sets. Pairs and Products. Relations.

Appendix B. Functions.

Functions and Maps. Composition of Functions. Injections and Surjections. Sequences and Families.

Appendix C. Equivalence Relations.

Equivalence Relations. Partitions.

Appendix D. The Integers.

Natural Numbers: Working Assumptions. Natural Numbers: Peano Postulates. Integers: Working Assumptions. Integers: Construction. Rational and Real Numbers.

Index.