INTRODUCTION TO MATHEMATICAL STRUCTURES AND PROOFS

This textbook is intended for a one term course whose goal is to ease the transition from lower division calculus courses, to upper level courses in algebra, analysis, number theory and so on. Without such a "bridge course", most instructors in advanced courses feel the need to start their courses with a review of the rudiments of logic, set theory, equivalence relations, and other basic mathematics before getting to the subject at hand. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve what we call "mathematical maturity", in other words, to develop an ability to understand and create mathematical proofs. Part of this transition involves learning to use the language of mathematics. This text spends a good deal of time exploring the judicious use of notation and terminology, and in guiding students to write up their solutions in clear and efficient language. Because this is an introductory text, the author makes every effort to give students a broad view of the subject, including a wide range of examples and imagery to motivate the material and to enhance the underlying intuitions.