A First Course in Abstract Mathematics

Learning Abstract mathematics is a slow and complex process. This is partly because a student of mathematics must not only try to grasp the ideas and master the techniques of approaching mathematical problems, but she must also learn the language of mathematics. This is often a cause of unease for many students in their first serious mathematics course. The present text intends to help such students. It is designed to be used as a textbook in introductory abstract mathematics courses that are usually taken by beginning undergraduate students of mathematics as well as undergraduate and graduate students of natural sciences, engineering, and economics who plan to study more advanced subjects in abstract mathematics. It can also be used as a self-study book for any student with little or no prior experience with abstract mathematics. The only prerequisite is a basic knowledge of high school algebra and elementary arithmetic. The book starts with a general introduction to the methodology of mathematics and its structural similarities and differences with natural sciences. It develops the foundations of elementary logic, treats various theorem types and proof methods, gives an extensive discussion of sets, relations, functions and cardinal numbers, and ends with a survey of some of the most central mathematical theories. This is intended to provide the beginning student with a global view of mathematics and a road map for future studies. In writing this class-room-tested textbook, a special effort has been made to focus on the most basic concepts and their development. In particular, unnecessarily extensive discussions and redundant examples have been avoided so that the main points are not lost.