Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares readers for the more abstract mathematics courses that follow calculus. This text introduces readers to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.
Communicating Mathematics; Sets; Logic; Direct Proof and Proof by Contrapositive; More on Direct Proof and Proof by Contrapositive; Existence and Proof by Contradiction; Mathematical Induction; Prove or Disprove; Equivalence Relations; Functions; Cardinalities of Sets; Proofs in Number Theory; Proofs in Calculus; Proofs in Group Theory; Proofs in Ring Theory (Online); Proofs in Linear Algebra (Online); Proofs in Topology (Online)
ÂFor all readers interested in mathematical proofs.
Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction' and `proof' has never been adequately explained (although Kriesel, Goodman and Martin-Lof have attempted axiomatisations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting arithmetic and constructive analysis are given.
The lecture courses in this work are derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles; put together in this book they form an invaluable introduction to proof theory that is aimed at both mathematicians and computer scientists.