An Invitation to Abstract Mathematics Preface to Instructors Preface to Students Acknowledgments I Whats Mathematics 1 Lets Play a Game! 2 Whats the Name of the Game 3 How to Make a Statement 4 Whats True in Mathematics 5 Famous Classical Theorems 6 Recent Progress in Mathematics II How to Solve It 7 Lets be Logical! 8 Setting Examples 9 Quantifier Mechanics 10 Mathematical Structures 11 Working in the Fields (and Other Structures) 12 Universal Proofs 13 The Domino Effect 14 More Domino Games 15 Existential Proofs 16 A Cornucopia of Famous Problems III Advanced Math for Beginners 17 Good Relations 18 Order Please! 19 Lets be Functional! 20 Now Thats the Limit! 21 Sizing It Up 22 Infinite Delights 23 Number Systems Systematically 24 Games Are Valuable! IV. Appendices A. Famous Conjectures in Mathematics B The Foundations of Set Theory C All Games Considered D Top 40 List of Math Theorems. Index. 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.