An Introduction to Mathematical Proofs.

Format:

Book

Author/Creator:

Loehr, Nicholas A.

Series:

Textbooks in mathematics (Boca Raton, Fla.)

Textbooks in Mathematics

Language:

English

Subjects (All):

Proof theory.

Physical Description:

1 online resource (413 pages).

Edition:

1st ed.

Place of Publication:

Milton : CRC Press LLC, 2019.

Summary:

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.

Contents:

<P>Logic</P><P>Propositions; Logical Connectives; Truth Tables</P><P>Logical Equivalence; IF-Statements</P><P>IF, IFF, Tautologies, and Contradictions</P><P>Tautologies; Quantifiers; Universes</P><P>Properties of Quantifiers: Useful Denials</P><P>Denial Practice; Uniqueness</P><P></P><B><P>Proofs</P></B><P>Definitions, Axioms, Theorems, and Proofs</P><P>Proving Existence Statements and IF Statements</P><P>Contrapositive Proofs; IFF Proofs</P><P>Proofs by Contradiction; OR Proofs</P><P>Proof by Cases; Disproofs</P><P>Proving Universal Statements; Multiple Quantifiers</P><P>More Quantifier Properties and Proofs (Optional)</P><P></P><B><P>Sets</P></B><P>Set Operations; Subset Proofs</P><P>More Subset Proofs; Set Equality Proofs</P><P>More Set Quality Proofs; Circle Proofs; Chain Proofs</P><P>Small Sets; Power Sets; Contrasting ? and ⁶</P><P>Ordered Pairs; Product Sets</P><P>General Unions and Intersections</P><P>Axiomatic Set Theory (Optional)</P><P></P><B><P>Integers</P></B><P>Recursive Definitions; Proofs by Induction</P><P>Induction Starting Anywhere: Backwards Induction</P><P>Strong Induction</P><P>Prime Numbers; Division with Remainder</P><P>Greatest Common Divisors; Euclid's GCD Algorithm</P><P>More on GCDs; Uniqueness of Prime Factorizations</P><P>Consequences of Prime Factorization (Optional)</P><P></P><B><P>Relations and Functions</P></B><P>Relations; Images of Sets under Relations</P><P>Inverses, Identity, and Composition of Relations</P><P>Properties of Relations</P><P>Definition of Functions</P><P>Examples of Functions; Proving Equality of Functions</P><P>Composition, Restriction, and Gluing</P><P>Direct Images and Preimages</P><P>Injective, Surjective, and Bijective Functions</P><P>Inverse Functions</P><P></P><B><P>Equivalence Relations and Partial Orders</P></B><P>Reflexive, Symmetric, and Transitive Relations</P><P>Equivalence Relations</P><P>Equivalence Classes</P><P>Set Partitions</P><P>Partially Ordered Sets</P><P>Equivalence Relations and Algebraic Structures (Optional)</P><P></P><B><P>Cardinality</P></B><P>Finite Sets</P><P>Countably Infinite Sets</P><P>Countable Sets</P><P>Uncountable Sets</P><P></P><B><P>Real Numbers (Optional)</P></B><P>Axioms for R; Properties of Addition</P><P>Algebraic Properties of Real Numbers</P><P>Natural Numbers, Integers, and Rational Numbers</P><P>Ordering, Absolute Value, and Distance</P><P>Greatest Elements, Least Upper Bounds, and Completeness</P><P></P><B><P>Suggestions for Further Reading</P></B>

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-000-70980-9

1-000-70962-0

0-429-32258-5

9780429322587

OCLC:

1128466687