Understanding Mathematical Proof

Content: Introduction The need for proof The language of mathematics Reasoning Deductive reasoning and truth Example proofs Logic and Reasoning Introduction Propositions, connectives, and truth tables Logical equivalence and logical implication Predicates and quantification Logical reasoning Sets and Functions Introduction Sets and membership Operations on sets The Cartesian product Functions and composite functions Properties of functions The Structure of Mathematical Proofs Introduction Some proofs dissected An informal framework for proofs Direct proof A more formal framework Finding Proofs Direct proof route maps Examples from sets and functions Examples from algebra Examples from analysis Direct Proof: Variations Introduction Proof using the contrapositive Proof of biconditional statements Proof of conjunctions Proof by contradiction Further examples Existence and Uniqueness Introduction Constructive existence proofs Non-constructive existence proofs Counter-examples Uniqueness proofs Mathematical Induction Introduction Proof by induction Variations on proof by induction Hints and Solutions to Selected Exercises Bibliography Index