# Introduction to Modern Mathematics

## 1st Edition

**Authors:**Helena Rasiowa

**eBook ISBN:**9781483274720

**Imprint:**North Holland

**Published Date:**1st January 1973

**Page Count:**352

## Description

Introduction to Modern Mathematics focuses on the operations, principles, and methodologies involved in modern mathematics.

The monograph first tackles the algebra of sets, natural numbers, and functions. Discussions focus on groups of transformations, composition of functions, an axiomatic approach to natural numbers, intersection of sets, axioms of the algebra of sets, fields of sets, prepositional functions of one variable, and difference of sets. The text then takes a look at generalized unions and intersections of sets, Cartesian products of sets, and equivalence relations.

The book ponders on powers of sets, ordered sets, and linearly ordered sets. Topics include isomorphism of linearly ordered sets, dense linear ordering, maximal and minimal elements, quasi-ordering relations, inequalities for cardinal numbers, sets of the power of the continuum, and Cantor's theorem. The manuscript then examines elementary concepts of abstract algebras, functional calculus and its applications in mathematical proofs, and propositional calculus and its applications in mathematical proofs.

The publication is a valuable reference for mathematicians and researchers interested in modern mathematics.

## Table of Contents

Foreword

Chapter 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 Theories

Chapter II. Natural Numbers. Proofs by Induction

1. An Axiomatic Approach to Natural Numbers. The Principle of Induction

2. Examples of Proofs by Induction

Chapter III. Functions

1. The Concept of Function

2. One-to-One Functions. Inverse Function

3. Composition of Functions

4. Groups of Transformations

Chapter IV. Generalized Unions and Intersections of Sets

1. The Concept of Generalized Union and Intersection

2. The Properties of Generalized Unions and Intersections of Sets

Chapter 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 Relations

Chapter 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 Function

Chapter 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 Numbers

Chapter 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 Theorem

Chapter 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 Sets

Chapter X. Linearly Ordered Sets

1. Linear Orderings

2. Isomorphism of Linearly Ordered Sets

3. Dense Linear Orderings

4. Continuous Linear Orderings

Chapter 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 Hypothesis

Chapter 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 Calculus

Chapter 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 Theories

Chapter 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 Algebras

List of Important Symbols

Author Index

Subject Index

## Details

- No. of pages:
- 352

- Language:
- English

- Copyright:
- © North Holland 1973

- Published:
- 1st January 1973

- Imprint:
- North Holland

- eBook ISBN:
- 9781483274720