Logic and Discrete Mathematics

"This is a very well-written brief introduction to discrete mathematics that emphasizes logic and set theory and has shorter sections on number theory, combinatorics, and graph theory." (MAA Reviews, 4 January 2016)

Willem Conradie, University of Johannesburg, South Africa. Valentin Goranko, University of Stockholm, Sweden.

List of Boxes xiii Preface xvii Acknowledgements xxi About the Companion Website xxiii 1. Preliminaries 1 1.1 Sets 2 1.1.1 Exercises 7 1.2 Basics of logical connectives and expressions 9 1.2.1 Propositions, logical connectives, truth tables, tautologies 9 1.2.2 Individual variables and quantifiers 12 1.2.3 Exercises 15 1.3 Mathematical induction 17 1.3.1 Exercises 18 2. Sets, Relations, Orders 20 2.1 Set inclusions and equalities 21 2.1.1 Properties of the set theoretic operations 22 2.1.2 Exercises 26 2.2 Functions 28 2.2.1 Functions and their inverses 28 2.2.2 Composition of mappings 31 2.2.3 Exercises 33 2.3 Binary relations and operations on them 35 2.3.1 Binary relations 35 2.3.2 Matrix and graphical representations of relations on finite sets 38 2.3.3 Boolean operations on binary relations 39 2.3.4 Inverse and composition of relations 41 2.3.5 Exercises 42 2.4 Special binary relations 45 2.4.1 Properties of binary relations 45 2.4.2 Functions as relations 47 2.4.3 Reflexive, symmetric and transitive closures of a relation 47 2.4.4 Exercises 49 2.5 Equivalence relations and partitions 51 2.5.1 Equivalence relations 51 2.5.2 Quotient sets and partitions 53 2.5.3 The kernel equivalence of a mapping 56 2.5.4 Exercises 57 2.6 Ordered sets 59 2.6.1 Pre-orders and partial orders 59 2.6.2 Graphical representing posets: Hasse diagrams 61 2.6.3 Lower and upper bounds. Minimal and maximal elements 63 2.6.4 Well-ordered sets 65 2.6.5 Exercises 67 2.7 An introduction to cardinality 69 2.7.1 Equinumerosity and cardinality 69 2.7.2 Exercises 73 2.8 Isomorphisms of ordered sets. Ordinal numbers 75 2.8.1 Exercises 79 2.9 Application: relational databases 80 2.9.1 Exercises 86 3. Propositional Logic 89 3.1 Propositions, logical connectives, truth tables, tautologies 90 3.1.1 Propositions and propositional connectives. Truth tables 90 3.1.2 Some remarks on the meaning of the connectives 90 3.1.3 Propositional formulae 91 3.1.4 Construction and parsing tree of a propositional formula 92 3.1.5 Truth tables of propositional formulae 93 3.1.6 Tautologies 95 3.1.7 A better idea: search for a falsifying truth assignment 96 3.1.8 Exercises 97 3.2 Propositional logical consequence. Valid and invalid propositional inferences 101 3.2.1 Propositional logical consequence 101 3.2.2 Logically sound rules of propositional inference. Logically correct propositional arguments 104 3.2.3 Fallacies of the implication 106 3.2.4 Exercises 107 3.3 The concept and use of deductive systems 109 3.4 Semantic tableaux 113 3.4.1 Exercises 117 3.5 Logical equivalences. Negating propositional formulae 121 3.5.1 Logically equivalent propositional formulae 121 3.5.2 Some important equivalences 123 3.5.3 Exercises 124 3.6 Normal forms. Propositional resolution 126 3.6.1 Conjunctive and disjunctive normal forms of propositional formulae 126 3.6.2 Clausal form. Clausal resolution 129 3.6.3 Resolution-based derivations 130 3.6.4 Optimizing the method of resolution 131 3.6.5 Exercises 132 4. First-Order Logic 135 4.1 Basic concepts of first-order logic 136 4.1.1 First-order structures 136 4.1.2 First-order languages 138 4.1.3 Terms and formulae 139 4.1.4 The semantics of first-order logic: an informal outline 143 4.1.5 Translating first-order formulae to natural language 146 4.1.6 Exercises 147 4.2 The formal semantics of firstorder logic 152 4.2.1 Interpretations 152 4.2.2 Variable assignment and term evaluation 153 4.2.3 Truth evaluation games 156 4.2.4 Exercises 159 4.3 The language of first-order logic: a deeper look 161 4.3.1 Translations from natural language into first-order languages 161 4.3.2 Restricted quantification 163 4.3.3 Free and bound variables. Scope of a quantifier 164 4.3.4 Renaming of a bound variable in a formula. Clean formulae 165 4.3.5 Substitution of a term fo