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.
By:
Nicholas A. Loehr
Imprint: CRC Press
Country of Publication: United Kingdom
Dimensions:
Height: 254mm,
Width: 178mm,
Weight: 916g
ISBN: 9780367338237
ISBN 10: 0367338238
Series: Textbooks in Mathematics
Pages: 412
Publication Date: 11 October 2019
Audience:
College/higher education
,
Primary
Format: Hardback
Publisher's Status: Active
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 Algorithm More on GCDs; Uniqueness of Prime Factorizations Consequences of Prime Factorization (Optional) Relations and Functions Relations; Images of Sets under Relations Inverses, Identity, and Composition of Relations Properties of Relations Definition of Functions Examples of Functions; Proving Equality of Functions Composition, Restriction, and Gluing Direct Images and Preimages Injective, Surjective, and Bijective Functions Inverse Functions Equivalence Relations and Partial Orders Reflexive, Symmetric, and Transitive Relations Equivalence Relations Equivalence Classes Set Partitions Partially Ordered Sets Equivalence Relations and Algebraic Structures (Optional) Cardinality Finite Sets Countably Infinite Sets Countable Sets Uncountable Sets Real Numbers (Optional) Axioms for R; Properties of Addition Algebraic Properties of Real Numbers Natural Numbers, Integers, and Rational Numbers Ordering, Absolute Value, and Distance Greatest Elements, Least Upper Bounds, and Completeness Suggestions for Further Reading
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.
