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
Copyright 1996, 340 pp.
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Transition to Advanced Mathematics-Mathematics
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.
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
- 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.
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
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.
Ordinary Induction. Recursion. Summation. Wellordering
and Strong Induction. Binomial Coefficients.
2. Number Theory.
The Division Theorem. Divisibility. Prime Numbers. Congruence.
3. The Real Numbers.
Fields. Ordered Fields. Nested Intervals and Completeness.
Isomorphism of Fields. Null Sequences and Limits. The Complex Number
Finite and Infinite Sets. Countable and 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.