Many students have trouble the first time they take a mathematics course in which proofs play a significant role This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read & write proofs The book begins with the basic concepts of logic & set theory to familiarize students with the language of mathematics & how it is interpreted These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs The author shows how complex proofs are built up from these smaller steps using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers relations functions & infinite sets To give students the opportunity to construct their own proofs this new edition contains over 200 new exercises selected solutions & an introduction to Proof Designer software No background beyond standard high school mathematics is assumed This book will be useful to anyone interested in logic & proofs computer scientists philosophers linguists & of course mathematicians