An Introduction to Mathematical Proofs: 1st Edition (Hardback) book cover

An Introduction to Mathematical Proofs

1st Edition

By Nicholas A. Loehr

CRC Press

416 pages | 67 B/W Illus.

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Hardback: 9780367338237
pub: 2019-10-28
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This book contains an introduction to mathematical proofs. The topics appearing in Parts I through VII constitute the standard core material in proofs course. Part VIII develops properties of the real numbers from the ordered field axioms. The book maintains a targeted focus on helping students master key skills needed for later work, as opposed to giving a dry treatment of logic and set theory. A friendly conversational style couples with the necessary level of precision and rigor. The lecture format facilitates a continual cycle of examples, summaries, and review of previous material. Every lecture ends with an immediate review of the main points just covered. Three review lectures give detailed summaries. The essential core material is supplemented by more advanced topics in optional sections. Heavy emphasis is placed on proof templates, creating proof outlines for complex statements based on the logical structure. Many sample proofs are accompanied by annotations. Our coverage of induction is more extensive than some other texts. A careful distinction between the graph of a function and the function itself is made.

Key Features:

  • Arranged by fifty one-hour lectures. The lecture format facilitates a continual cycle of examples, summaries, and

    review of previous material.

  • Parts I and VII cover all the essential topics for a Transition to Advanced Mathematics course.
  • Part VIII offers advanced topics typcially found in an Advanced Calculus course.
  • Heavy emphasis is placed on proof templates, which create proof outlines for complex statements.
  • Induction is covered more than in other texts.

Table of Contents


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


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)


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)


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)


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

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.

About the Series

Textbooks in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Mathematical Analysis