Proofs and Fundamentals

In an effort to make advanced mathematics accessible to a wide variety of students, and to give even the most mathematically inclined students a solid basis upon which to build their continuing study of mathematics, there has been a tendency in recent years to introduce students to the for­ mulation and writing of rigorous mathematical proofs, and to teach topics such as sets, functions, relations and countability, in a "transition" course, rather than in traditional courses such as linear algebra. A transition course functions as a bridge between computational courses such as Calculus, and more theoretical courses such as linear algebra and abstract algebra. This text contains core topics that I believe any transition course should cover, as well as some optional material intended to give the instructor some flexibility in designing a course. The presentation is straightforward and focuses on the essentials, without being too elementary, too exces­ sively pedagogical, and too full to distractions. Some of features of this text are the following: (1) Symbolic logic and the use of logical notation are kept to a minimum. We discuss only what is absolutely necessary - as is the case in most advanced mathematics courses that are not focused on logic per se.