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A First Course in Mathematical Logic and Set Theory by Michael L. O'Leary

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CONTENTS

Preface

Acknowledgments

List of Symbols

1 Propositional Logic

1.1 Symbolic Logic

Propositions

Propositional Forms

Interpreting Propositional Forms

Valuations and Truth Tables

1.2 Inference

Semantics

Syntactics

1.3 Replacement

Semantics

Syntactics

1.4 Proof Methods

Deduction Theorem

Direct Proof

Indirect Proof

1.5 The Three Properties

Consistency

Soundness

Completeness

2 First-Order Logic

2.1 Languages

Predicates

Alphabets

Terms

Formulas

2.2 Substitution

Terms

Free Variables

Formulas

2.3 Syntactics

Quantifier Negation

Proofs with Universal Formulas

Proofs with Existential Formulas

2.4 Proof Methods

Universal Proofs

Existential Proofs

Multiple Quantifiers

Counterexamples

Direct Proof

Existence and Uniqueness

Indirect Proof

Biconditional Proof

Proof of Disunctions

Proof by Cases

3 Set Theory

3.1 Sets and Elements

Rosters

Famous Sets

Abstraction

3.2 Set Operations

Union and Intersection

Set Difference

Cartesian Products

Order of Operations

3.3 Sets within Sets

Subsets

Equality

3.4 Families of Sets

Power Set

Union and Intersection

Disjoint and Pairwise Disjoint

4 Relations and Functions

4.1 Relations

Composition

Inverses

4.2 Equivalence Relations

Equivalence Classes

Partitions

4.3 Partial Orders

Bounds

Comparable and Compatible Elements

Well-Ordered Sets

4.4 Functions

Equality

Composition

Restrictions and Extensions

Binary Operations

4.5 Injections and Surjections

Injections

Surjections

Bijections

Order Isomorphims

4.6 Images and Inverse Images

5 Axiomatic Set Theory

5.1 Axioms ...

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