See our review of the second edition. For the fourth edition, the author has added two new chapters on groups, going as far as Lagrange’s Theorem and its applications. These chapters are intended to serve as an introduction to abstract algebra. There is also new material on inequalities, counting methods, the inclusion-exclusion principle, and Euler’s \(\phi\) function.

These changes put the book firmly in the position of a “transition’ or “bridge’ course, assuming that students have taken calculus and linear algebra and preparing them for the next steps. Liebeck’s book stands out from the crowd of similar books by being short (as the title says, it is concise) and by trying to expose students to mathematical ideas beyond the basics of sets and logic. In addition to the pre-Analysis and pre-Algebra chapters, there are chapters on complex numbers, inequalities, some number theory and combinatorics, and the Platonic solids. Students are taught how to understand and create proofs, but they are also given a glimpse of what it is all for.

My one regret is the title. It is already very easy for students whose interest is mostly in applied mathematics to conclude that all the proofs in their Algebra and Analysis courses are of no value. To label the transition course an introduction to “pure mathematics’ is to reinforce that feeling. I suspect that if I were asked to teach this course, that problem would be enough to keep me from adopting the book. Which is a pity. After all, applied mathematicians also need to know about proofs, counting, inequalities, bounds, and even groups — and this book could help them learn all that.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.