Introduction to Discrete Mathematics with ISETL

Intended for first- or second-year undergraduates, this introduction to discrete mathematics covers the usual topics of such a course, but applies constructivist principles that promote - indeed, require - active participation by the student. Working with the programming language ISETL, whose syntax is close to that of standard mathematical language, the student constructs the concepts in her or his mind as a result of constructing them on the computer in the syntax of ISETL. This dramatically different approach allows students to attempt to discover concepts in a "Socratic" dialog with the computer. The discussion avoids the formal "definition-theorem" approach and promotes active involvement by the reader by its questioning style. An instructor using this text can expect a lively class whose students develop a deep conceptual understanding rather than simply manipulative skills. Topics covered in this book include: the propositional calculus, operations on sets, basic counting methods, predicate calculus, relations, graphs, functions, and mathematical induction.

1 Numbers and Programs.- 1.1 The Basics of ISETL.- Activities.- Discussion.- Beginning with ISETL.- Some Syntax.- Familiar Sets of Numbers.- Decimal Representation.- Binary Representation.- Sequences.- Exercises.- 1.2 Divisibility.- Activities.- Discussion.- ISETL funcs—Functions.- ISETL smaps—Functions.- Sources of Functions.- Recursive Functions.- Modular Arithmetic.- Prime Numbers.- Common Divisors.- Common Multiples.- Exercises.- Overview of Chapter 1.- 2 Propositional Calculus.- 2.1 Boolean Expressions.- Activities.- Discussion.- Constants and Variables.- Basic Operations.- Functions Using Boolean Values.- Exercises.- 2.2 Implication and Proof.- Activities.- Discussion.- Conditional Statements.- Variations of Conditional Statements.- Direct Proof.- Indirect Proof.- Proof by Contradiction.- Exercises.- Overview of Chapter 2.- 3 Sets and Tuples.- 3.1 Defining Sets and Tuples.- Activities.- Discussion.- Sets and their Elements.- Tuples and their Elements.- Forming Sets and Tuples.- Sequences.- Recursive Sequences.- Exercises.- 3.2 Operations on Sets.- Activities.- Discussion.- Cardinality.- Subsets.- Basic Combinations of Sets.- De Morgan’s Laws.- Cartesian Products.- Inclusion-Exclusion.- Exercises.- 3.3 Counting Methods.- Activities.- Discussion.- The Multiplication Principle.- Permutations.- Combinations.- The Pigeonhole Principle.- Exercises.- Overview of Chapter 3.- 4 Predicate Calculus.- 4.1 Quantified Expressions.- Activities.- Discussion.- Existential and Universal Quantifiers.- Quantifying over Proposition Valued Functions—Existential.- Quantifying over Proposition Valued Functions—Universal.- Negations.- Reasoning about Quantified Expressions.- Exercises.- 4.2 Multi-Level Quantification.- Activities.- Discussion.- Quantified Statements that Depend on a Variable.- Two-Level Quantification.- Negating Two-Level Quantifications.- Reasoning about Two-Level Quantifications.- Three-Level Quantification.- Exercises.- Overview of Chapter 4.- 5 Relations and Graphs.- 5.1 Relations and their Graphs.- Activities.- Discussion.- Relations.- Representing a Relation.- Properties of Relations.- More about Graphs.- Exercises.- 5.2 Equivalence Relations and Graph Theory.- Activities.- Discussion.- Equivalence Relations.- Types of Graphs.- Subgraphs.- Planarity.- Exercises.- Overview of Chapter 5.- 6 Functions.- 6.1 Representing Functions.- Activities.- Discussion.- Constructing Functions.- Functions as Expressions.- Functions as Sequences.- Functions as Tables.- Functions as Graphs.- The Process of a Function.- Two Definitions.- Exercises.- 6.2 Properties of Functions.- Activities.- Discussion.- Basic Properties.- One-to-One Functions.- Combinations of Functions.- Inverse Functions.- Rate of Growth for Functions.- Exercises.- Overview of Chapter 6.- 7 Mathematical Induction.- 7.1 Understanding the Method.- Activities.- Discussion.- Proposition-Valued Functions.- Eventually Constant Proposition-Valued Functions.- Implication-Valued Functions.- Modus Ponens.- Coordinating the Steps.- Exercises.- 7.2 Using Mathematical Induction.- Activities.- Discussion.- Making Induction Proofs.- The Induction Principle.- Complete Induction.- The Binomial theorem.- Exercises.- Overview of Chapter 7.- 8 Partial Orders.- Activities.- Discussion.- Order on a Set.- Diagrams of Posets.- Topological Sorting.- Sperner’s Theorem.- Exercises.- Overview of Chapter 8.- 9 Infinite Sets.- Discussion.- Sets of Equal Cardinality.- Infinite Sets.- Countable Sets.- Uncountable Sets.- Ordering of Infinite Sets.- Exercises.- Appendix 1: Getting Started With Isetl.- A. Working in the Execution Window.- B. Working with Files.- C. Using Directives.- D. Graphing in ISETL.- Appendix 2: Some Special Code.- Index of Frequently Used Sets and Functions.