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May 28 , 2009

Basic Language of Mathematics


This book originates as an essential underlying component of a modern, imaginative three-semester honors program (six undergraduate courses) in Mathematical Studies. In its entirety, it covers Algebra, Geometry and Analysis in One Variable.

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The book is intended to provide a comprehensive and rigorous account of the concepts of set, mapping, family, order, number (both natural and real), as well as such distinct procedures as proof by induction and recursive definition, and the interaction between these ideas; with attempts at including insightful notes on historic and cultural settings and information on alternative presentations. The work ends with an excursion on infinite sets, principally a discussion of the mathematics of Axiom of Choice and often very useful equivalent statements.

  • Sets
  • Mappings
  • Properties of Mappings
  • Families
  • Relations
  • Ordered Sets
  • Completely Ordered Sets
  • Induction and Recursion
  • The Natural Numbers
  • Finite Sets
  • Finite Sums
  • Countable Sets
  • Some Algebraic Structures
  • The Real Numbers: Complete Ordered Fields
  • The Real Number System
  • The Real Numbers: Existence
  • Infinite Sets

Readership: Undergraduate and graduate students in mathematics; Mathematicians.
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