A First Course in Abstract Mathematics

Ali Mustafazade


CATEGORY: Mathematics
PAGES: 279
SIZE: 19.5 x 25 cm.
EDITION: 1st print, October 2013
HARDCOVER ISBN: 9786055250225

Learning abstract mathematics is a slow and complex process. This is partly because a student of mathematics must not only try to grasp the ideas and master the techniques of approaching mathematical problems, but she must also learn the language of mathematics. This is often a cause of unease for many students in their first serious mathematics course. The present text intends to help such students. It is designed to be used as a textbook in introductory abstract mathematics courses that are usually taken by beginning undergraduate students of mathematics as well as undergraduate and graduate students of natural sciences, engineering, and economics who plan to study more advanced subjects in abstract mathematics. It can also be used as a self-study book for any student with little or no prior experience with abstract mathematics. The only prerequisite is a basic knowledge of high school algebra and elementary arithmetic.

The book starts with a general introduction to the methodology of mathematics and its structural similarities and differences with natural sciences. It develops the foundations of elementary logic, treats various theorem types and proof methods, gives an extensive discussion of sets, relations, functions and cardinal numbers, and ends with a survey of some of the most central mathematical theories. This is intended to provide the beginning student with a global view of mathematics and a road map for future studies.

In writing this class-room-tested textbook, a special effort has been made to focus on the most basic concepts and their development. In particular, unnecessarily extensive discussions and redundant examples have been avoided so that the main points are not lost.

Ali Mostafazadeh received his PhD degree in Physics, from University of Texas at Austin. Currently, he is teaching at Koç University, Department of Mathematics.


Preface vii

1 Introduction

1.1 General Structure of Mathematics

1.2 Relationship with Natural Sciences

1.3 Mathematical Language

2 Elements of Logic

2.1 Statements and Predicates

2.2 Qualifiers

2.3 Negation

2.4 Compound Statements

2.5 Contradictions and Tautologies

2.6 Propositional Calculus

Comments and Suggestions for Further Reading


3 Theorem Types and Proof Methods

3.1 Existence Theorems

3.2 Uniqueness Theorems

3.3 Classification Theorems

3.4 Proving Implications

3.4.1 Trivial Proof

3.4.2 Direct Proof

3.4.3 Contrapositive Proof

3.4.4 Deductive Proof

3.5 Proof by Contradiction

3.6 Proof by Cases

3.7 Proof by Induction

3.7.1 Induction

3.7.2 Complete Induction

3.7.3 Recursive Definitions

3.8 Characterization Theorems

3.9 Examples and Counterexamples

Comments and Suggestions for Further Reading


4 Sets

4.1 Elementary Properties of Sets

4.2 Intersection of Sets

4.3 Union of Sets

4.4 Cartesian Product of Sets

4.5 Set of Natural Numbers

Comments and Suggestions for Further Reading


5 Relations

5.1 Basic Properties of Relations

5.2 Composition and Inverse of Relations

5.3 Equivalence Relations

5.4 Ordering Relations

5.4.1 Partial and Total Ordering Relations

5.4.2 Graphical Representation of Ordering Relations

5.4.3 Least and Greatest Elements, Supremum and Infimum

5.4.4 Well-Ordering

Comments and Suggestions for Further Reading


6 Functions

6.1 Basic Properties of Functions

6.2 One-to-One and Onto Functions

6.3 Composition of Functions

6.4 Inverse Function

6.5 Functions acting in {1, 2, 3, · · · , n}

6.6 Sequences

Comments and Suggestions for Further Reading


7 Cardinal Numbers

7.1 Equivalent Sets

7.2 Finite Sets

7.3 Infinite Sets

7.4 Countable Sets

7.5 Uncountable Sets

7.6 Cardinality

7.7 Continuum Hypothesis

7.8 Axiom of Choice

Comments and Suggestions for Further Reading


8 Mathematical Theories

8.1 Mathematical Structures and Invariants

8.2 Posets

8.3 Groups

8.4 N, Z, Q, and R as Mathematical Structures

8.5 Metric Spaces

8.6 Topological Spaces

Comments and Suggestions for Further Reading


Appendix A Total Ordering of N

Appendix B Constructing Z and Q

Appendix C Power Set of a Finite Set

Appendix D Countable Union of Countable Sets

Appendix E Cantor-Schr¨oder-Bernstein Theorem



This book is the Turkish translation of The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford, 2002. […]

AUTHOR(S) Roger Penrose, 42TL

The Emperor’s New Mind
Concerning Computers, Minds, and the Laws of Physics

AUTHOR(S) Roger Penrose,