Understanding Mathematical Proof

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students ability to understand proofs and construct correct proofs of their own.The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples.

Captivate your audience and enhance your storytelling with this tutorial based 4-color cookbook, featuring dozens of solutions to your titling needs. Each chapter includes case studies and interviews with the pros, lending cutting insight and lessons learned that will have you creating inspired title sequences in no time.The book features genre-based tutorial sections, with step by step instructions for creating effective horror, comedy, drama, and suspense titling sequences. Tutorials for creating some of the most popular title sequences in blockbuster movies are included (Se7en, The Sopranos, 24, The Matrix). Other tutorials teach you how to effectively use sound and VFX in your titles, and also included is instruction on editing your title sequence. These techniques, as well as chapters on the essentials of typography allow you to apply these lessons to your title sequence regardless of whether it''s for TV, the web, or digital signage.Also included is a DVD with sample clips, as well as project files that allow you to refine the techniques you learned in the book. As an added bonus we''ve included 3 titling chapters from other Focal books, with specific instructions on titling within certain software applications.Cover images provided by MK12, from The Alphabet Conspiracy. Learn more at www.MK12.com