An Introduction to Mathematical Proofs

An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No prerequisites are needed beyond high-school algebra. New material is presented in small chunks that are easy for beginners to digest. The author offers a friendly style without sacrificing mathematical rigor. Ideas are developed through motivating examples, precise definitions, carefully stated theorems, clear proofs, and a continual review of preceding topics. Features Study aids including section summaries and over 1100 exercises Careful coverage of individual proof-writing skills Proof annotations and structural outlines clarify tricky steps in proofs Thorough treatment of multiple quantifiers and their role in proofs Unified explanation of recursive definitions and induction proofs, with applications to greatest common divisors and prime factorizations About the Author: Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra. Logic Propositions; Logical Connectives; Truth Tables Logical Equivalence; IF-Statements IF, IFF, Tautologies, and Contradictions Tautologies; Quantifiers; Universes Properties of Quantifiers: Useful Denials Denial Practice; Uniqueness Proofs Definitions, Axioms, Theorems, and Proofs Proving Existence Statements and IF Statements Contrapositive Proofs; IFF Proofs Proofs by Contradiction; OR Proofs Proof by Cases; Disproofs Proving Universal Statements; Multiple Quantifiers More Quantifier Properties and Proofs (Optional) Sets Set Operations; Subset Proofs More Subset Proofs; Set Equality Proofs More Set Quality Proofs; Circle Proofs; Chain Proofs Small Sets; Power Sets; Contrasting ∈ and ⊆ Ordered Pairs; Product Sets General Unions and Intersections Axiomatic Set Theory (Optional) Integers Recursive Definitions; Proofs by Induction Induction Starting Anywhere: Backwards Induction Strong Induction Prime Numbers; Division with Remainder Greatest Common Divisors; Euclid’s GCD Algorith