Discrete Mathematics. A Concise Introduction

Preface
	What to Include? What to Omit?
	The Chapters
Contents
1 Some Elementary Informal Set Theory
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	1.1 Russell\'s ``Paradox\'\'
2 Safe Set Theory
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		2.1 The ``Real Sets\'\'
		2.2 What Caused Russell\'s Paradox
		2.3 Some Useful Sets
		2.4 Operations on Classes and Sets
		2.5 The Powerset
		2.6 The Ordered Pair and Finite Sequences
		2.7 The Cartesian Product
			2.7.1 Strings or Expressions Over an Alphabet
		2.8 Exercises
3 Relations and Functions
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	3.1 Relations
	3.2 Transitive Closure
		3.2.1 Computing the Transitive Closure
		3.2.2 The Special Cases of Reflexive Relations on Finite Sets
		3.2.3 Warshall\'s Algorithm
	3.3 Equivalence Relations
	3.4 Partial Orders
	3.5 Functions
		3.5.1 Lambda Notation
		3.5.2 Kleene Extended Equality for Function Calls
		3.5.3 Function Composition
	3.6 Finite and Infinite Sets
	3.7 Diagonalisation and Uncountable Sets
	3.8 Operators and the Cantor-Bernstein Theorem
		3.8.1 An Application of Operators to Cardinality
	3.9 Exercises
4 A Tiny Bit of Informal Logic
	4.1 Enriching Our Proofs to Manipulate Quantifiers
	4.2 Exercises
5 Induction
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	5.1 Inductiveness Condition (IC)
	5.2 IC Over mathbbN
		5.2.1 Well-Foundedness
		5.2.2 Induction Examples
	5.3 Inductive Definitions of Functions
		5.3.1 Examples on Inductive Function Definitions
		5.3.2 Fibonacci-like Inductive Definitions; Course-of-Values Recursion
	5.4 Exercises
6 Inductively Defined Sets; Structural Induction
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	6.1 Set Closures
	6.2 Induction Over a Closure
	6.3 Closure Versus Definition by Stages
	6.4 Exercises
7 Recurrence Equations and Their Closed-Form Solutions
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	7.1 Big-O, Small-o, and the ``Other\'\' sim
	7.2 Solving Recurrences; the Additive Case
	7.3 Solving Recurrences; the Multiplicative Case
	7.4 Generating Functions
	7.5 Exercises
8 An Addendum to Trees
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	8.1 Trees: More Terminology
	8.2 A Few Provable Facts About Trees
	8.3 An Application to Summations
	8.4 How Many Trees?
	8.5 Exercises
References
Index