Introduction to Discrete Mathematics with ISETL

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.