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.

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.
Contents:

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.