An Introduction to Mathematical Proofs Textbooks in Mathematics Series

An Introduction to Mathematical Proofspresents 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

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.