Volume 2: Counting (Rigorous Elementary Mathematics)

In mathematical contests and olympiads, there are four subject areas: algebra, counting, number theory, and geometry. These areas are addressed in the four books of the Rigorous Elementary Mathematics series. While there are many books that are geared towards helping students to learn how to solve problems on contests, those books tend to assume prerequisite knowledge without proofs, and they instead focus on applying theorems to problems. This 4-volume series fills the gap in the literature by proving many of the relevant theorems in a logically sequenced framework from the ground up. Throughout this series, three ideas are reinforced: writing the same object in multiple ways, breaking up equality using antisymmetry, and using equivalence relations. It is suggested that the books be read by those who have experience with mathematical proofs and problem-solving. The ideal audience consists of teachers of mathematics who want to solidify their own fundamentals, and outstanding students who want a second look at the material. Volume 2: Counting contains various counting principles, including bijections, casework, complementary counting, combinatorial multiplication and division, and inclusion-exclusion. For proving identities, double counting and algebraic methods are presented. Special topics include variations of the pigeonhole principle, ways of distributing balls across boxes, and graph theory with Ramsey theory. The final chapters are about probability, including the probabilistic method, recursive sequences like Fibonacci and Catalan and linear recurrences, and basic group theory. The key insight of this volume is the understanding that combinatorics is about finding the cardinality of finite sets.