
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
 
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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.
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.
