Introduction to Modern Mathematics

ForewordChapter I. The Algebra of Sets 1. The Concept of Set 2. The Union of Sets 3. The Intersection of Sets. The Laws of Absorption and Distributivity 4. The Difference of Sets. Relationships Between the Difference of Sets and the Operations of Union and Intersection of Sets 5. The Universe. The Complement of a Set 6. Axioms of the Algebra of Sets 7. Fields of Sets 8. Propositional Functions of One Variable 9. Note on Axioms of Set Theory 10. Comments on the Need of an Axiomatic Approach to Set Theory and on Axiomatic TheoriesChapter II. Natural Numbers. Proofs by Induction 1. An Axiomatic Approach to Natural Numbers. The Principle of Induction 2. Examples of Proofs by InductionChapter III. Functions 1. The Concept of Function 2. One-to-One Functions. Inverse Function 3. Composition of Functions 4. Groups of TransformationsChapter IV. Generalized Unions and Intersections of Sets 1. The Concept of Generalized Union and Intersection 2. The Properties of Generalized Unions and Intersections of SetsChapter V. Cartesian Products of Sets. Relations. Functions as Relations 1. Cartesian Products 2. Binary Relations 3. Propositional Functions of Two Variables 4. Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, and Transitive Relations 5. Functions as RelationsChapter VI. Generalized Products, m-Ary Relations. Functions of Several Variables. Images and Inverse Images Under a Function 1. Generalized Products 2. m-Ary Relations 3. Propositional Functions of m Variables 4. Functions of Several Variables 5. Images and Inverse Images Under a FunctionChapter VII. Equivalence Relations 1. Definition of Equivalence Relations. Method of Identification 2. Application of the Method of Identification to the Construction of Integers 3. Application of the Method of Identification to the Construction of Rational Numbers 4. Note on Cantor's Theory of Real NumbersChapter VIII. Powers of Sets 1. Equipotent Sets. Power of a Set 2. Enumerable Sets 3. Examples of Non-Enumerable Sets 4. Inequalities for Cardinal Numbers. The Cantor-Bernstein Theorem 5. Sets of the Power of the Continuum 6. The Power Set. Cantor's Theorem. Consequences of Cantor's TheoremChapter IX. Ordered Sets 1. Ordering Relations 2. Maximal and Minimal Elements 3. Subsets of Ordered Sets. The Kuratowski-Zorn Lemma 4. Note on Lattices 5. Quasi-Ordering Relations 6. Note on Directed SetsChapter X. Linearly Ordered Sets 1. Linear Orderings 2. Isomorphism of Linearly Ordered Sets 3. Dense Linear Orderings 4. Continuous Linear OrderingsChapter XI. Well-Ordered Sets 1. Well-Ordering Relations. Ordinal Numbers 2. Comparison of Ordinal numbers 3. Sets of Ordinal Numbers 4. Powers of Ordinal Numbers. The Cardinal Number N(𝔪) 5. The Theorem on Transfinite Induction. Transfinite Sequences 6. The Theorem on Definition by Transfinite Induction 7. Zermelo's Well-Ordering Theorem. Note on the Axiom of Choice 8. A Proof of the Kuratowski-Zorn Lemma 9. The Continuum HypothesisChapter XII. The Propositional Calculus and its Applications in Mathematical Proofs 1. Introductory Remarks 2. Propositional Connectives 3. The Concept of Law in the Propositional Calculus 4. The Concept of Rules of Inference. The Rule of Detachment 5. Equivalence of Propositions and Equivalence of Propositional Functions 6. The Rules of Detachment for Equivalence 7. Square of Opposition 8. The Rules of Hypothetical Syllogism 9. Rules of Inference Involving Conjunction and Disjunction 10. Simplification, Frege's, Duns Scotus', and Clavius' Rules 11. Apagogic Proofs 12. Principal Propositional Tautologies and their Applications 13. Axiomatic Approach to the Propositional CalculusChapter XIII. The Functional Calculus and its Applications in Mathematical Proofs 1. Quantifiers and Propositional Functions of One Variable 2. Quantifiers with Restricted Ranges 3. Quantifiers and Propositional Functions of m Variables 4. Functional Tautologies 5. Introduction and Elimination Laws for Quantifiers 6. Distributivity Laws for Quantifiers 7. Laws on the Relettering and on the Alternations of Quantifiers 8. Rules of Inference 9. Quantifiers Versus Generalized Unions and Intersections of Sets 10. Examples of Applications of the Functional Calculus in Mathematical Proofs 11. Note on Formalized Mathematical TheoriesChapter XIV. Elementary Concepts of Abstract Algebra 1. Abstract Algebras 2. Subalgebras. Sets of Generators 3. Similar Algebras. Homomorphisms. Isomorphisms 4. Congruences. Quotient Algebras 5. Products of Algebras 6. Algebraic Functions 7. Equationally Definable Classes of Algebras 8. Free Algebras 9. Construction of Free Algebras for Certain Classes of AlgebrasList of Important SymbolsAuthor IndexSubject Index