Introduction to Mathematical Structures and Proofs: (Textbooks in Mathematical Sciences)

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. The exercise sets range from routine exercises, to more thoughtful and challenges ones.

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plug and chug"), how to build a solution, and ultimately he wants to teach the students to take a statement and develop a way to prove it...Gerstein writes with a certain flair that I think students will find appealing. For instance, after his discussion of cardinals he has a section entitled Languages and Finite Automata. This allows him to illustrate some of the ideas he has been discussing with problems that almost anyone can understand, but most importantly he shows how these rather transparent problems can be subjected to a mathematical analysis. His discussion of how a machine might determine whether the sequence of words "Celui fromage de la parce que maintenant" is a legitimate French sentence is just delightful (and even more so if one knows a little French.)...I am confident that a student who works through Gerstein's book will really come away with (i) some mathematical technique, and (ii) some mathematical knowledge.- Steven Krantz, American Mathematical Monthly"This very elementary book is intended to be a textbook for a one-term course which introduces students into the basic notions of any higher mathematics courses...The explanations of the basic notions are combined with some main theorems, illustrated by examples (with solutions if necessary) and complemented by exercises. The book is well written and should be easily understandable to any beginning student."- S. Gottwald, ZentralblattThis textbook is intended for a one-term course whose goal is to ease the transition from lower-division calculus courses to upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, combinatorics, etc. It contains a wide-ranging assortment of examples and imagery to motivate and to enhance the underlying intuitions, as well as numerous exercises and a solutions manual for professors.