This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

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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 and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and 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, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and 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 and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

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On December 12, 1937, Japanese aircraft sank the American gunboat Panay, which was anchored in the Yangtze River outside Nanjing, China. Although the Japanese apologized, the attack turned American public opinion against Japan, and President Roosevelt dispatched Captain Royal Ingersoll to London to begin conversations with the British admiralty about Japanese aggression in the Far East. While few Americans remember the Panay Incident, it established the first links in the chain of Anglo-American military collaboration that eventually triumphed in World War II. In The Origins of the Grand Alliance, William T. Johnsen provides the first comprehensive analysis of military collaboration between the United States and Great Britain before the Second World War. He sets the stage by examining Anglo-French and Anglo-American coalition military planning from 1900 through World War I and the interwar years. Johnsen also considers the formulation of policy and grand strategy, operational planning, and the creation of the command structure and channels of communication. He addresses vitally important logistical and materiel issues, particularly the difficulties of war production. Military conflicts in the early twenty-first century continue to underscore the increasing importance of coalition warfare for historian and soldier alike. Drawn from extensive sources and private papers held in the United Kingdom, Canada, and the United States, Johnsen's exhaustively researched study refutes the idea that America was the naive junior partner in the coalition and casts new light on the US-UK "special relationship."