An Invitation to Abstract Mathematics (Undergraduate Texts in Mathematics) - Tapa dura

From the reviews:

“Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... Bajnok’s book can most certainly be used as a text for a traditional transition course designed for mathematics majors. He wrote the book with a broader audience in mind, and personally uses it at Gettysburg College for anyone interested in deepening his or her mathematical knowledge ... .” (Jill Dietz, MAA Reviews, May, 2014)

“This book can be added to the growing population of ‘transition’ books--texts aimed at bridging the leap from the traditional calculus sequence to upper-division coursework in mathematics. The topics typically associated with such a text are all present: basic logic, predicates and quantifiers, induction, relations, and functions. ... any reader able to complete this work and its problems will be suitably prepared for further studies in mathematics. Summing Up: Recommended. Lower- and upper-division undergraduates.” (D. S. Larson, Choice, Vol. 51 (5), January, 2014)

“This textbook aims for the spot in many departments’ curriculum where students are introduced to advanced mathematics. ... the author introduces the game in an applied setting that the reader will likely find both compelling and intriguing: a set of options for two corporations in competition. ... This text definitely focuses on mathematics. The material covers a wide range of material, probably more than most instructors will cover in one semester ... .” (John Perry, zbMATH, Vol. 1274, 2013)

This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind.  The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse.  The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader.  With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible.  This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.

Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.