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2013 | Book

Let’s Play a Game! Read chapter Read first chapter

An Invitation to Abstract Mathematics

Author: Béla Bajnok

Publisher: Springer New York

Book Series : Undergraduate Texts in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind. The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse. The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader. With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible. This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.

Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.

Table of Contents

Frontmatter

What's Mathematics

Chapter 1. Let’s Play a Game!
Abstract
We start our friendship with abstract mathematics with an example that illuminates how concrete quantitative problems may turn into abstract mathematical situations and how mathematicians might be led from specific questions to the development of highly abstract concepts and discoveries.
Béla Bajnok
Chapter 2. What’s the Name of the Game?
Abstract
Abstract mathematics deals with the analysis of mathematical concepts and statements. Mathematics is unique among all disciplines in that its concepts have a precise and consistent meaning, and its results, once established, are not subject to opinions or experimental verification and remain valid independently of time, place, and culture—although their perceived importance might vary. In this chapter we discuss mathematical concepts; in Chap. 3 we study mathematical statements.
Béla Bajnok
Chapter 3. How to Make a Statement?
Abstract
In the previous chapter we learned how to introduce mathematical concepts with definitions or as primitives. Once we introduce a new concept, we are interested in its properties, usually stated as mathematical statements. Statements are sentences that are either true or false—but not both.
Béla Bajnok
Chapter 4. What’s True in Mathematics?
Abstract
In the last chapter we made the assertions that the statements
  • The number \({2}^{n-1}({2}^{n} - 1)\) is a perfect number for each positive integer n for which 2 n − 1 is a prime number.
  • If 2 n − 1 is a prime number for some positive integer n, then n is a prime number.
are true, and we promised arguments that demonstrate them without any doubt. We will provide these in this chapter.
Béla Bajnok
Chapter 5. Famous Classical Theorems
Abstract
There are many famous theorems in mathematics. Some are known for their importance, others for their depth, usefulness, or sheer beauty. In this chapter we discuss seven of the most remarkable classical theorems; in the next chapter, we discuss three others from more recent times. Our choices for this top ten list were motivated primarily by the nature of their proofs; we apologize if we did not choose your favorite theorem. (A more representative top 40 list can be found in Appendix D at the end of the book.) Here we included theorems that are considered to have the oldest, the most well-known, the most surprising, the most elegant, and the most unsettling proofs. Some of the theorems in our list were disappointing—even angering—to mathematicians of the time, others were celebrated instantly by most.
Béla Bajnok
Chapter 6. Recent Progress in Mathematics
Abstract
In the last chapter we discussed seven of the most famous classical theorems of mathematics. We now turn to three more recent results to complete our top ten list. We will not provide any proofs—in fact, there are very (very!) few people who have seen complete proofs for these results.
Béla Bajnok

How to Solve It?

Frontmatter
Chapter 7. Let’s Be Logical!
Abstract
In arithmetic we learned how to perform arithmetical operations (addition, multiplication, taking negatives, etc.) on numbers; then, in algebra, we generalized arithmetic using variables instead of numbers. Similarly, we can build compounded statements from simple statements, and we can study their general structures. The branch of mathematics dealing with the structure of statements is called logic. A study of the rules of logic is essential when one studies correct reasoning.
Béla Bajnok
Chapter 8. Setting Examples
Abstract
When we are interested in studying several objects at the same time, we may put these objects into a set. Some of the most commonly used sets in mathematics are the following sets of numbers:
Béla Bajnok
Chapter 9. Quantifier Mechanics
Abstract
We introduced abstract mathematics in Chap. 1 with Hackenbush games; in particular, we analyzed one such game, the Aerion. Here we review this rather simple game as it will enable us to discuss quantifiers in a simple and natural way. Consider the following figure:
Béla Bajnok
Chapter 10. Mathematical Structures
Abstract
In Chap. 8 we made the somewhat heuristic claim that properties and identities about statements can easily be altered so that they also hold true for sets. As an example, we considered the claim that
$$\displaystyle{(P \vee \neg Q \vee R) \wedge (\neg (P \vee R)) \Leftrightarrow \neg P \wedge \neg Q \wedge \neg R}$$
holds for all statements P, Q, and R.
Béla Bajnok
Chapter 11. Working in the Fields (and Other Structures)
Abstract
In Chap. 10 we got acquainted with mathematical structures such as groups, rings, integral domains, fields, and Boolean algebras. As discussed there, the benefit of abstracting the common properties of various systems into a unifying structure is that, once we prove certain statements about a structure using only the properties that apply to all models of the structure, they will then be true for each system that models the structure. In this chapter we see examples for such axiomatic proofs.
Béla Bajnok
Chapter 12. Universal Proofs
Abstract
We have already discussed the role of proofs in mathematics and have seen a variety of examples for proofs (e.g., in Chaps. 4, 5, and 11). Having learned about logic, sets, and quantifiers, we are now able to study proofs more formally and thus deepen our understanding of them.
Béla Bajnok
Chapter 13. The Domino Effect
Abstract
In Chap. 12 we studied universal statements of the form
$$\displaystyle{\forall a \in U,P(a)}$$
for given sets U and predicates P. Here we continue this discussion by examining the case when U is the set of natural numbers.
Béla Bajnok
Chapter 14. More Domino Games
Abstract
In this chapter we discuss three variations of induction: strong induction, split induction, and double induction. This arsenal of induction techniques will allow us to prove a variety of fundamental and powerful mathematical statements.
Béla Bajnok
Chapter 15. Existential Proofs
Abstract
In the last several chapters we discussed proof techniques for universal statements of the form
$$\displaystyle{\forall x \in U,P(x);}$$
in this chapter we focus on the existential quantifier and analyze existential statements of the form
$$\displaystyle{\exists x \in U,P(x).}$$
For instance, we may claim that a certain equation has a real number solution (the existence of \(\sqrt{2}\), to be formally proven only in Chap.​ 23, is a prime example), or we may claim that a certain set has a minimum element (by Theorem 13.6, every nonempty set of natural numbers does). Quite often, we deal with statements of the form
$$\displaystyle{\forall x \in U,\exists y \in V,P(x,y);}$$
for example, when in Chap.​ 1 we claimed that a certain game had a winning strategy for Player 2, we made an existential statement that for any sequence of moves by Player 1, there was a response by Player 2 that resulted in a win for Player 2.
Béla Bajnok
Chapter 16. A Cornucopia of Famous Problems
Abstract
Just like we have many famous theorems in mathematics, we have many famous problems. (In fact, there really is no clear distinction between theorems and solved problems or, similarly, between conjectures and unsolved problems.) In Chaps.5 and 6 we discussed our top ten list of most famous classical theorems; in this chapter we feature a list of ten problems whose solutions, while considered elementary—that is, not requiring any knowledge beyond what we already have at this point—provide challenges beyond the typical problems of the earlier chapters.
Béla Bajnok

Advanced Math for Beginners

Frontmatter
Chapter 17. Good Relations
Abstract
We begin our adventure into advanced mathematics with the study of one of the most fundamental objects: relations.
Béla Bajnok
Chapter 18. Order, Please!
Abstract
As promised, in this chapter we discuss an important and highly applicable type of relations: order relations. Since the usual “less than or equal to” relation on ℝ (or a subset of ℝ such as ℕ, ℤ, or ℚ) is a primary example of an order relation, it is useful to use notation resembling the≤sign; however, since our discussion applies to other orderings as well, we choose the symbol≼that is similar, but not identical, to the symbol≤. Throughout this chapter (and beyond), a≼b denotes the fact that a∼ R b holds for some elements a and b (of a given set) and order relation R (on the same set).
Béla Bajnok
Chapter 19. Let’s Be Functional!
Abstract
In this chapter we discuss functions. Although the concept of functions is undoubtedly familiar, here we follow a more abstract approach; in particular, we consider functions as special relations.
Béla Bajnok
Chapter 20. Now That’s the Limit!
Abstract
We have seen examples for infinite sequences throughout this book; in Chap. 19 we finally defined them officially as functions whose domain is the set of natural numbers ℕ or, equivalently, as the elements of the infinite Cartesian product X ∞ for some set X. In this chapter we study the most important attribute of some sequences: their limits.
Béla Bajnok
Chapter 21. Sizing It Up
Abstract
Counting is probably one of our earliest intellectual pursuits, and it is a ubiquitous task in everyday life. The principles of counting are also what several branches of mathematics are based on, especially combinatorics, probability theory, and statistics.
Béla Bajnok
Chapter 22. Infinite Delights
Abstract
In Chap. 20 we defined what it means for a sequence to have an infinite limit. In this chapter we discuss a different aspect of the intriguing concept of infinity; namely, as promised in Chap. 21, we study sets of infinite size. As we will soon see, not all infinite sets are created equal: some are “larger” (“much larger”) than others.
Béla Bajnok
Chapter 23. Number Systems Systematically
Abstract
Throughout this book, we frequently considered our familiar sets of numbers:
ℕ: the set of natural numbers
ℤ: the set of integers
ℚ: the set of rational numbers
ℝ: the set of real numbers
ℂ: the set of complex numbers
Béla Bajnok
Chapter 24. Games Are Valuable!
Abstract
In this chapter we return to our very first adventure in this book: the analysis of games. As an illustrative example, we evaluate our good old game Aerion of Chap.​ 1; namely, we show that it has exactly a “one half move advantage” for player A (we will, of course, make this notion precise).
Béla Bajnok
Backmatter
Metadata
Title
An Invitation to Abstract Mathematics
Author
Béla Bajnok
Copyright Year
2013
Publisher
Springer New York
Electronic ISBN
978-1-4614-6636-9
Print ISBN
978-1-4614-6635-2
DOI
https://doi.org/10.1007/978-1-4614-6636-9

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