An Introduction to the Language of Mathematics

This is a textbook for an undergraduate mathematics major transition course from technique-based mathematics (such as Algebra and Calculus) to proof-based mathematics. It motivates the introduction of the formal language of logic and set theory and develops the basics with examples, exercises with solutions and exercises without. It then moves to a discussion of proof structure and basic proof techniques, including proofs by induction with extensive examples. An in-depth treatment of relations, particularly equivalence and order relations completes the exposition of the basic language of mathematics. The last chapter treats infinite cardinalities. An appendix gives some complement on induction and order, and another provides full solutions of the in-text exercises. 

The primary audience is undergraduate mathematics major, but independent readers interested in mathematics can also use the book for self-study.

Chapter 1- The language of logic and set-theory.- Chapter 2- On proofs and writing mathematics.- Chapter 3- Relations.- Chapter 4- Cardinality.- Appendix A- Complements.- Appendix B- Solutions to exercises in the text.- Index.- Bibliography.

Frédéric Mynard is currently a professor of mathematics at New Jersey City University. Originally from France where he obtained his PhD in 1999, he has been teaching at various universities in the US since 2001, teaching a wide spectrum of mathematics courses, from entry level to graduate courses. A topologist, he has published 35 peer-reviewed research papers, and the book Convergence Foundations of Topology, revisiting classical General Topology from the point of view of his area of research: convergence spaces.