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

**HARDCOVER PRICE:** 25 TL

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

Problems

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

Problems

**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

Problems

**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

Problems

**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

Problems

**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

Problems

**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

Problems

**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**

**Index**

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

Roger Penrose, 42The Emperor’s New Mind

Concerning Computers, Minds, and the Laws of Physics