Learning Discrete Mathematics with Isetl - Baxter

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Résumé :
The title of this book, Learning Discrete Mathematics with ISETL raises two issues. We have chosen the word Learning rather than Teaching because we think that what the student does in order to learn is much more important than what the professor does in order to teach. Academia is filled with outstanding mathematics teachers: excellent expositors, good organizers, hard workers, men and women who have a deep understanding of Mathematics and its applications. Yet, when it comes to ideas in Mathe? matics, our students do not seem to be learning. It may be that something more is needed and we have tried to construct a book that might provide a different kind of help to the student in acquiring some of the fundamental concepts of Mathematics. In a number of ways we have made choices that seem to us to be the best for learning, even if they don't always completely agree with standard teaching practice. A second issue concerns students' writing programs. ISETL is a pro? gramming language and by the phrase with ISETL in the title, we mean that our intention is for students to write code, think about what they have written, predict its results, and run their programs to check their predic? tions. There is a trade-off here. On the one hand, it can be argued that students' active involvement with constructing Mathematics for themselves and solving problems is essential to understanding concepts.

Sommaire:
1 Numbers, Programs, and ISETL.- 1.1 Preview.- 1.2 Overview of ISETL.- 1.3 Integers.- 1.4 Integer Operations.- 1.5 Special Binary Integer Operators.- 1.5.1 Extending Binary Operators.- 1.5.2 Greatest Common Divisor.- 1.5.3 Modular Arithmetic.- 1.6 Random Integers.- 1.7 Floating-Point, Rational, and Real Numbers.- 1.8 Floating-Point Operations.- 1.9 Other Simple Data Types.- 2 Propositional Calculus.- 2.1 Preview.- 2.2 Boolean Variables and Expressions.- 2.3 Reasoning and Boolean Expressions.- 2.4 Methods of Proof.- 2.5 Predicate Calculus: First Pass.- 2.6 Propositions as Objects.- 3 Sets and Tuples.- 3.1 Preview.- 3.2 Introduction to Sets.- 3.3 Set Operations.- 3.4 Set Formation.- 3.5 Tuples and Tuple Operations.- 3.6 One-Liners.- 3.6.1 Proofs.- 3.6.2 Prime Numbers.- 3.7 The Compound Operator.- 4 Functions.- 4.1 Preview.- 4.2 Representations of Functions.- 4.2.1 Programs.- 4.2.2 Funcs.- 4.2.3 Tuples and Strings.- 4.2.4 Smaps.- 4.3 Function Operations.- 4.3.1 Evaluation and Equality of Functions.- 4.3.2 Images and Pre-images.- 4.3.3 Arithmetic with Functions.- 4.3.4 Composition of Functions.- 4.3.5 The Inverse of a Function.- 4.3.6 The Partial Sum Operator.- 4.3.7 Collections of Functions.- 4.3.8 Functions of Several Variables.- 4.4 Applications of Functions.- 4.4.1 Data Bases.- 4.4.2 Counting Onto Functions.- 4.4.3 Finite State Automata.- 5 Predicate Calculus.- 5.1 Preview.- 5.2 Single-Level Quantification.- 5.3 Two-Level Quantification.- 5.4 Three-Level Quantification.- 5.5 Quantification in Math and CS.- 5.5.1 Continuity of Functions.- 5.5.2 The Chemical Factory Problem.- 5.5.3 Abstract Algebra.- 5.5.4 Statements About Programs.- 6 Combinatorics, Matrices, Determinants.- 6.1 Preview.- 6.2 Combinatorics.- 6.2.1 The Multiplication and Addition Principles.- 6.2.2 Permutations.- 6.2.3 Combinations.- 6.3 Matrices.- 6.3.1 Representation of Matrices.- 6.3.2 Operations on Matrices.- 6.3.3 Matrix Inversion.- 6.4 Determinants.- 7 Mathematical Induction.- 7.1 Preview.- 7.2 Proposition Valued Functions of N.- 7.3 Modus Ponens.- 7.4 Implication Valued Functions of N.- 7.5 Making Proofs by Induction.- 7.6 Variations on the Induction Theme.- 7.6.1 Double Induction.- 7.6.2 Strong Induction.- 7.6.3 Recursively Defined Sequences.- 8 Relations and Graphs.- 8.1 Preview.- 8.2 Relations and Their Representations.- 8.3 Properties of Relations.- 8.4 Digraphs.- ISETL order form.