Discrete Mathematics via Logic and Proof

Description

This textbook introduces discrete mathematics by emphasizing the importance of reading and writing proofs. Because it begins by carefully establishing a familiarity with mathematical logic and proof, this approach suits not only a discrete mathematics course, but can also function as a transition to proof. Its unique, deductive perspective on mathematical logic provides students with the tools to more deeply understand mathematical methodology-an approach that the author has successfully classroom tested for decades. Chapters are helpfully organized so that, as they escalate in complexity, their underlying connections are easily identifiable. Mathematical logic and proofs are first introduced before moving onto more complex topics in discrete mathematics. Some of these topics include: Mathematical and structural induction Set theory Combinatorics Functions, relations, and ordered sets Boolean algebra and Boolean functions Graph theory Introduction to Discrete Mathematics via Logic and Proof will suit intermediate undergraduates majoring in mathematics, computer science, engineering, and related subjects with no formal prerequisites beyond a background in secondary mathematics. Preface Topics Selected Intended Audiences Goals and Approach Prerequisites and Course Emphases For Students: Reading a Mathematics Text Acknowledgements List of Notations Logical Acronyms Contents 1 Propositional Logic 1.1 A Gentle Introduction to Logic and Proof 1.2 Conjunction, Disjunction, and Negation 1.3 Argument Semantics for Propositional Logic 1.4 Conditional and Biconditional Sentences 1.5 Introduction to Deduction; Rules for AND 1.6 Elimination Rules for CONDITIONALS 1.7 Introduction Rules for CONDITIONALS 1.8 Proof by Contradiction: Rules for NOT 1.9 Inference Rules for OR 2 First-Order Logic 2.1 Symbolizing Sentences 2.2 First-Order Logic: Syntax and Semantics 2.3 Rules for Identity and Universal Quantifiers 2.4 Rules for Existential Quantifiers 3 Mathematical Induction and Arithmetic 3.1 Mathematical Induction and Recursion 3.2 Variations on Mathematical Induction and Recursion 3.3 Recurrence Relations; Structural Induction 3.4 Peano Arithmetic 3.5 Divisibility 4 Basic Set Theory and Combinatorics 4.1 Relations and Operations on Sets 4.2 Collections of Sets and the Power Set 4.3 Multiplicative Counting Principles 4.4 Combinations 4.5 Additive Counting Principles 5 Set Theory and Infinity 5.1 Countably Infinite Sets 5.2 Uncountably Infinite Sets 5.3 Formal Set Theory and the Halting Problem 6 Functions and Equivalence Relations 6.1 Functions and Their Properties 6.2 Composite Functions and Inverse Functions 6.3 Equivalence Relations and Partitions 6.4 The Integers and Modular Arithmetic 7 Posets, Lattices, and Boolean Algebra 7.1 Partially Ordered Sets 7.2 Lattices 7.3 From Boolean Lattices to Boolean Algebra 7.4 Boolean Functions and Logic Circuits 7.5 Representing Boolean Functions 7.6 Simplifying Boolean Functions 8 Topics in Graph Theory 8.1 Eulerian Trails 8.2 Hamiltonian Paths 8.3 Planar Graphs 8.4 Coloring Graphs *-21ptImage Credits A Inference Rules for PL and FOL *-16ptIndex Index