This book introduces the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader to understand and construct proofs and write clear mathematics. The authors achieve this by exploring set theory, combinatorics and number theory, which include many fundamental mathematical ideas. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all time great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.