Elementary Point-Set Topology: A Transition to Advanced Mathematics
256
Elementary Point-Set Topology: A Transition to Advanced Mathematics
256eBook
Available on Compatible NOOK devices, the free NOOK App and in My Digital Library.
Related collections and offers
Overview
Starting with elementary concepts in logic and basic techniques of proof writing, the text defines topological and metric spaces and surveys continuity and homeomorphism. Additional subjects include product spaces, connectedness, and compactness. The final chapter illustrates topology's use in other branches of mathematics with proofs of the fundamental theorem of algebra and of Picard's existence theorem for differential equations.
"This is a back-to-basics introductory text in point-set topology that can double as a transition to proofs course. The writing is very clear, not too concise or too wordy. Each section of the book ends with a large number of exercises. The optional first chapter covers set theory and proof methods; if the students already know this material you can start with Chapter 2 to present a straight topology course, otherwise the book can be used as an introduction to proofs course also." — Mathematical Association of America
Product Details
| ISBN-13: | 9780486811017 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/10/2016 |
| Series: | Aurora: Dover Modern Math Originals |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 33 MB |
| Note: | This product may take a few minutes to download. |
About the Author
André L. Yandl is Professor Emeritus of Mathematics at Seattle University.
Adam Bowers is a Lecturer in Mathematics at the University of California, San Diego.
Read an Excerpt
Elementary Point-Set Topology
A Transition to Advanced Mathematics
By André L. Yandl, Adam Bowers
Dover Publications, Inc.
Copyright © 2016 André L. Yandl and Adam BowersAll rights reserved.
ISBN: 978-0-486-81101-7
CHAPTER 1
Mathematical Proofs and Sets
1.1 Introduction to Elementary Logic
In everyday language, terms are defined using other terms, which inevitably leads to circular definitions. In mathematics, we begin with some terms that are undefined terms. Using these, we then introduce defined terms. Using undefined and defined terms, we make statements that are universally accepted as either true or false. Such statements are called propositions. We must stress the fact that a statement may be a proposition even though it may not be possible to verify whether it is true or false. For example, the statement "President John F. Kennedy drank exactly three glasses of wine on his twenty first birthday" is either true or false, hence a proposition; however, we may never be able to determine if it is true or if it is false.
In a mathematical theory, propositions that are accepted as true without proof are called axioms or postulates. The truth of all other propositions must be proved using the axioms as a start and rules of logic that have been carefully stated. Propositions that have to be proved true are called theorems. Theorems that are stated and proved as preliminaries for the main theorem are called lemmas. Theorems that follow from the main theorem are called corollaries. In mathematics, once we accept the truth of the original assumptions (the axioms), we must accept the truth of all theorems that are derived from them. Research mathematicians state propositions they suspect to be true. These are called conjectures. When the truth of a conjecture has been demonstrated, the conjecture becomes a theorem. However, if someone succeeds in demonstrating one situation in which the conjecture is not true, then this situation is called a counterexample to the conjecture, and the conjecture is regarded as false.
In this section, and the next two, we introduce the rules of logic and methods of proof that will be followed in proving theorems. Some propositions are simple in the sense that they do not contain other propositions as components. However, new propositions can be formed using simple propositions and logical connectives. It is convenient to define compound propositions by displaying their truth values for all possible truth values of their components in tables called truth tables. We will see later that if a compound proposition has n components, then there are 2n possible combinations of truth values of the components. For example, if a proposition has 3 components, then there are 8 possible combinations of truth values for these components. (See Example 1.1.5.)
Definitions 1.1.1. Given the propositions p and q, the conjunction of p and q (denoted p [conjunction] q), the disjunction of p and q (denoted p [disjunction] q), and the negation of p (denoted [logical not] p) are defined by the truth tables below.
A conjunction is intended to express "and," whereas a disjunction is used to express "or." (When we say "or," we use it to mean "inclusive or" as opposed to "exclusive or," because we allow both propositions to be true at the same time.) A negation is used to mean "not."
It is important to note that in the definitions above, p and q are variable propositions that have no truth values until p and q are replaced by propositions whose truth values are known.
Definition 1.1.2. A propositional form is an expression involving letters that represent simple propositions and a finite number of connectives such as [conjunction], [disjunction] [logical not], and others to be defined later.
Just as we do when we perform arithmetical operations with signed numbers, we avoid writing many parentheses by adhering to the following rule: First, [logical not] is applied to the simplest proposition following it. Second, [conjunction] connects the simplest propositions on each side of it. Third, [disjunction] connects the simplest propositions on each side of it. For example, the negation of "p [conjunction] q" must be written "[logical not] (p [conjunction] q" and not "[logical not] p [conjunction] q." However, "[([logical not] p) [conjunction] q] [disjunction] r" may be written "[logical not] p [conjunction] q [disjunction] r."
Example 1.1.3. Let p be the proposition "Normal cats have four legs" and let q be the proposition "Normal eagles have three wings." For each of the following, find the truth values:
(a)"Normal cats have four legs and normal eagles have three wings,"
(b)"Normal cats have four legs or normal eagles have three wings,"
(c)"It is not the case that normal cats have four legs."
Solution. Observe first that p is true and q is false.
(a)This proposition is the conjunction p [conjunction] q. It is false since one of the components is false.
(b)This proposition is the disjunction p [disjunction] q and is true since one of its components is true.
(c)This proposition is the negation [logical not] p. This proposition is false since it is the negation of a true statement.
Example 1.1.4. Is the proposition "The authors of this text play tennis poorly" the negation of the proposition "The authors of this text play tennis well?"
Solution. The answer is "No," since both propositions are false. The authors of this text do not play tennis at all.
We often encounter propositional forms with more than two simple components. In the next example, we will make up a truth table for a proposition with three simple components.
Example 1.1.5. Make a truth table for the proposition [logical not] p [conjunction] (q [disjunction] r).
Solution. This proposition has three simple components and two main components: [logical not] p and q [disjunction] r. In the truth table, we first determine the truth values of the main components, and then using the definition of conjunction, we fill in the column for [logical not] p [conjunction] (q [disjunction] r).
Definition 1.1.6. Two propositional forms are equivalent if and only if they have the same truth values for all possible combinations of truth values of their components.
Example 1.1.7. Show that the propositional forms [logical not] (p [conjunction] q) and [logical not] p [disjunction] q are equivalent.
Solution. Since each propositional form has the two simple components p and q, there are four possible combinations of truth values for p and q. In the truth table below, we show the truth values of (p [conjunction] q) and [logical not] p [disjunction] [logical not] q in Columns 4 and 7, respectively.
Observe that the same truth values appear on the same line of Columns 4 and 7. Thus the propositional forms [logical not] (p [conjunction] q) and [logical not] p [disjunction] [logical not] q are equivalent.
Exercises
1. Which of the following are propositions?
(a) Paris is the capital of France.
(b) The first author of this text ate pancakes on January 5, 1952.
(c) Where are my glasses?
(d) For all real numbers x, x2< 0.
(e)This statement is false.
2. Make a truth table for each of the following.
(a) p [disjunction] [logical not] p
(b) [logical not] (p [disjunction] q)
(c) p [disjunction] (q [conjunction] r)
(d) p [conjunction] (q [disjunction] r)
(e) [logical not] (p [conjunction] [logical not] q)
3. Determine which of the following pairs of propositional forms are equivalent.
(a) [logical not] (p [conjunction] q) and [logical not] p [conjunction] q
(b) [logical not] p [disjunction] q and [logical not] p [conjunction] q
(c) p [disjunction][logical not] q and [logical not] p [disjunction] q
(d) [logical not] p [disjunction] [logical not] q and [logical not] (p [conjunction] q)
(e) (p [disjunction] q) [disjunction]r and p [disjunction] (q [disjunction] r)
(f) (p [conjunction] q) [conjunction] r and p [conjunction] (q [conjunction] r)
(g) p [conjunction] (q [disjunction] r) and (p [conjunction] q) [disjunction] (p [conjunction] r)
(h) p [disjunction] (q [conjunction] r) and (p [disjunction] q) [conjunction] (p [disjunction] r)
(i) p [disjunction] ([logical not] q [conjunction] r) and ([logical not] p [disjunction] q) [conjunction] (p [disjunction] [logical not] r)
4. Suppose p and q denote true propositions and r and s denote false propositions. What is the truth value of each of the following?
(a) [logical not] p [disjunction] r [conjunction] q
(b) p [conjunction] q [disjunction] r [conjunction] s
(c)(p [disjunction] [logical not] q) [conjunction] ([logical not] r [disjunction] s)
(d)([logical not] r [disjunction] s) [conjunction] ([logical not] p [conjunction] q)
1.2 More Elementary Logic
The student likely will remember that most theorems stated and proved in elementary mathematics are of the form "If p, then q." In this propositional form, p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent). This propositional form is called a conditional proposition or an implication and is abbreviated p [??] q (read "p implies q"). The symbol [??] is called an implication symbol. We define the conditional proposition by means of a truth table.
Definition 1.2.1. Let p and q be propositions. The truth values for the conditional proposition p [??] q are given in the table below.
Observe that p [??] q is false only in the case where p is true and q is false.
Theorem 1.2.2.If p and q are propositions, then p [??] q, [logical not] p [disjunction] q, and [logical not] (p [conjunction] [logical not] q) are equivalent.
Proof. The truth values of the three propositions are displayed in Columns 3, 5, and 8 of the truth table below.
Observe how the entries of Columns 3, 5, and 8 are identical. Thus, the three propositions p [??] q, [logical not] p [disjunction] q, and [logical not](p [conjunction] [logical not] q) are equivalent. Q.E.D.
The symbol Q.E.D. that appears at the end of the proof abbreviates the Latin phrase quod erat demonstrandum, which means "what was to be demonstrated." (On occasion, students will jokingly claim that it abbreviates "quite easily done.")
There are three propositions that are closely related to p [??] q. We now introduce these propositions.
Definitions 1.2.3. Let p and q denote two propositions. The proposition q [??] p is the converse of p [??] q, the proposition [logical not] p [??] q is its inverse, and [logical not] q [??] [logical not] p is its contrapositive.
Theorem 1.2.4.Let p and q be propositions. The implication p [??] q and its contrapositive [logical not] q [??]p are equivalent.
Proof. The following truth table indicates the truth values for the implication p [??] q as well as its contrapositive, converse, and inverse.
Columns 3 and 6 are identical, and so p [??] q and [logical not] q [??] [logical not] p are equivalent. Q.E.D.
In the above truth table, Columns 7 and 8 both differ from Column 3. This shows that an implication is not equivalent to its converse or inverse. However, Columns 7 and 8 are identical, which means that the converse and inverse are equivalent to each other.
We have stated earlier that two propositional forms are equivalent if they have the same truth values for all possible combinations of truth values of their components. It is not surprising then, that when the propositions p and q are equivalent, we say p is true if and only if q is true. Formally, we have the following definition.
Definition 1.2.5. Let p and q be propositions. The propositional form p [??] q (read "p if and only if q") is true precisely when p and q have the same truth values, or when p and q are equivalent. We call this propositional form an equivalence or a double implication. The truth table for p [??] q is given below.
[ILLUSTRATION OMITTED]
Definition 1.2.6. A tautology is a propositional form that is true for all possible truth values of its components.
A simple example of a tautology is p [??] p, where p is any proposition. No matter the truth value of p, both sides of p [??] p will necessarily have the same truth value. Therefore, p [??] p is always true.
Example 1.2.7. Let p be a proposition. Show that p [??] [logical not] ([logical not] p) is a tautology.
Solution. We construct a truth table:
[ILLUSTRATION OMITTED]
Since the entries of Column 4 are all Ts, the propositional form is a tautology.
Because of Example 1.2.7, whenever we encounter [logical not] ([logical not] p) in an argument, we can replace it by the equivalent proposition p. As we shall soon see, when we are faced with the task of solving a problem, we must have the ability to replace propositions by equivalent propositions until we get a proposition that yields a solution to the problem.
Example 1.2.8. Solve the equation x2 + 12x = 13, where x is a real number.
Solution. When we write x2 + 12x = 13, it represents the proposition "x satisfies the equation x2 + 12x = 13." This proposition may be true or false depending on x. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The equation (x - 1)(x + 13) = 0 is satisfied when either factor equals zero, hence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, the equation is satisfied if x = 1 or x = -13.
The following theorem contains a number of pairs of equivalent propositional forms.
Theorem 1.2.9. Let p, q, and r denote propositions. Each of the following is a tautology:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Proof. The proof of (e) was given in Example 1.1.7. We prove (c) as an illustration and leave the proof of the other parts for the exercises. Consider the truth table below.
All entries in Columns 4 and 6 are identical. This means that the propositional forms p [??] [logical not] q and [logical not] (p [conjunction] q) have the same truth values for all combinations of truth values for p and q. Thus, all entries in Column 7 are Ts, and so the propositional form [p [??] [logical not] q] [??] [(p [conjunction] q)] is a tautology. Q.E.D.
(Continues...)
Excerpted from Elementary Point-Set Topology by André L. Yandl, Adam Bowers. Copyright © 2016 André L. Yandl and Adam Bowers. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Preface List of Figures List of Symbols1 Mathematical Proofs and Sets1.1 Introduction to Elementary Logic .1.2 More Elementary Logic 1.3 Quantifiers1.4 Methods ofMathematical Proof1.5 Introduction to Elementary Set Theory 1.6 Cardinality 1.7 Cardinal Arithmetic2 Topological Spaces
2.1 Introduction 2.2 Topologies2.3 Bases2.4 Subspaces 2.5 Interior, Closure, and Boundary2.6 Hausdorff spaces2.7 Metric Spaces 2.8 Euclidean Spaces3 Continuous Functions
3.1 Review of the Function Concept3.2 More on Image and Inverse Image3.3 Continuous Functions3.4 More on Continuous Functions3.5 More on Homeomorphism 4 Product Spaces
4.1 Products of Sets4.2 Product Spaces4.3 More on Product Spaces 5 Connectedness
5.1 Introduction to Connectedness5.2 Products of Connected Spaces5.3 Connected Subsets of the Real Line 6 Compactness
6.1 Introduction to Compactness 6.2 Compactness in the Space of Real Numbers6.3 The Product of Compact Spaces 6.4 Compactness in Metric Spaces6.5 More on Compactness in Metric Spaces 6.6 The Cantor Set7 Fixed Point Theorems and Applications
7.1 Sperner’s 7.2 Brouwer’s Fixed Point Theorem.7.3 The Fundametnal Theorem of Algebra 7.4 Function Spaces7.5 ContractionsIndex