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An Introduction to Mathematical Proofs: Textbooks in Mathematics

Autor Nicholas A. Loehr
en Limba Engleză Paperback – 21 ian 2023
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.
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Specificații

ISBN-13: 9781032475226
ISBN-10: 1032475226
Pagini: 412
Ilustrații: 67
Dimensiuni: 178 x 254 x 25 mm
Greutate: 0.73 kg
Ediția:1
Editura: CRC Press
Colecția CRC Press
Seria Textbooks in Mathematics


Public țintă

Undergraduate

Notă biografică

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.

Cuprins

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

Descriere

This book contains an introduction to mathematical proofs, including fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The book is divided into approximately fifty brief lectures. Each lecture corresponds rather closely to a single class meeting.