1st Edition

An Introduction to Mathematical Proofs

By Nicholas A. Loehr Copyright 2020
    412 Pages 67 B/W Illustrations
    by CRC Press

    412 Pages 67 B/W Illustrations
    by CRC Press

    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 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

    Biography

    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.