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Overview
Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter includes a problem set arranged in order of increasing difficulty as well as review exercises and annotated references suggesting sources for further study. In addition to three helpful Appendixes, the book concludes with answers and hints for selected problems.
Product Details
| ISBN-13: | 9780486796680 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 07/08/2014 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 400 |
| File size: | 24 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 18 Years |
Read an Excerpt
A First Course in Geometry
By Edward T. Walsh
Dover Publications, Inc.
Copyright © 2014 Edward T. WalshAll rights reserved.
ISBN: 978-0-486-79668-0
CHAPTER 1
THE LANGUAGE OF MATHEMATICS
1.1 INTRODUCTION
If your intuition about physical space is fairly well developed, many of the ideas developed in this book will seem quite familiar to you. For although the geometry presented here is not the only way to think about space (more will be said about this in Chapter 5), it coincides with the notions most of us come to take for granted as we are growing up. On the other hand, if your geometric intuition is not very good, your formal study of geometry here should improve the situation.
In order that we may proceed with some confidence, our approach will be systematic. Specifically, we will employ what is often called "the axiomatic method." This method was apparently invented by the Greeks of the fourth, fifth, and sixth centuries B.C. to securely systematize the geometry they were developing, in the face of some paradoxes they were unable to resolve.
The elements of the axiomatic method are few. We begin with some primitive or undefined terms, some definitions, and a set of non-contradictory assumptions called postulates or axioms.
We then endeavor to make plausible conjectures about relationships among objects of our study, employing the defined and undefined terms. Our attempts to establish the validity of these conjectures are referred to as proofs, and the valid conjectures are called theorems.
Since we want to discuss these conjectures with some measure of clarity and precision, it is obvious why we should first agree on the meanings of technical terms involved (definitions), as well as how these terms may be related to each other (axioms). But why must we also begin with some undefined terms? It would certainly be convenient to have a definition for every mathematical term that we encounter. But is this possible? Consider the following definitions from Webster's Collegiate Dictionary
happening,n. Occurrence.
occurrence,n. Appearance or happening.
appearance,n. The action or an instance of appearing; an occurrence.
Note that these definitions haven't really told us anything since each of the words is defined in terms of the others. Such sets of definitions are referred to as circular definitions, and although our trio is far less poetic, it is about as logically meaningful as Gertrude Stein's "A rose is a rose is a rose ..."
This problem of circularity arises whenever we attempt to define every term we use. To avoid this difficulty we will agree that certain terms will remain undefined, and, in particular, that definitions and other general statements (axioms, theorems, etc.) can be expressed using these primitive undefined terms. The meanings of the undefined terms will, however, become clearer as the subject unfolds.
In this book the undefined terms adopted are point, line, plane, and set. Our axioms, definitions, and theorems will employ these terms in a manner that is probably consistent with your view of the real world, for, after all, geometry has been studied and expanded primarily in order to provide men with a clearer vision of physical space. We will use the axiomatic method merely as a tool to guide us to this clearer vision. It will provide us with some measure of certainty that the conclusions we are coming to are not contradictory, and that they do follow logically from the geometric facts previously established (logic will be discussed at some length in the following pages). Some care must be taken not to let the formalism of the axiomatic method obscure the beauty of the geometrical structure.
1.2 LOGIC: PROPOSITIONS AND TRUTH TABLES
In the fourth century B.C., the Greek philosopher Aristotle systematized deductive logic, sometimes referred to as the art and science of reasoning. The models of argumentation shown to us by Aristotle in his development of formal logic are still acceptable to most literate Westerners. Moreover, despite periodic attacks on Aristotelian logic by writers of the past few centuries, to study logic was to study the formal logic of Aristotle until, in 1847, George Boole published a thin volume, entitled The Mathematical Analysis of Logic. In his book Boole, who was an English elementary school teacher at the time, attempted to give a mathematical formulation to Aristotelian logic. This mathematical mode of systematizing the "laws of thought" reached maturity in the years 1910–1913, just after the publication of the great Principia Mathematica, by Alfred North Whitehead and Bertrand Russell.
A brief study of some of the more elementary and fundamental aspects of mathematical logic should lead you to understand how to use some of the important rules of logic employed so often in mathematical arguments. To simplify our work and to free ourselves somewhat from the ambiguities of our language, we will use a symbolic language akin to the one you learned while studying the algebra of the real number system. We will call this algebra the algebra of propositions. A proposition is merely a declarative sentence that is unambiguous; for example:
2 is an even integer.
Some Californians are not wealthy.
Our president is a man of peace.
In your study of the algebra of the real number system you used variables to represent real numbers. In this second algebra, the variables will represent propositions. We will agree that these variables must take one (and only one!) of the two values true (T) and false (F). That is, a proposition must be either true or false, but not both. We will consider some special operations defined on propositions, for just as we combine numerical expressions algebraically through the operations of addition and multiplication, we combine sentences with operations known as conjunction, disjunction, implication, and equivalence.
In a conjunction we combine two given propositions by placing the word "and" between them. Consider the propositions
Geometry is interesting.
Logic is a game.
Their conjunction is as follows:
Geometry is interesting and logic is a game.
If we denote the first two propositions by p and q, then the proposition which is their conjunction is denoted p [conjunction] q, read "p and q." We will agree that a conjunction is true whenever both the sentences that make it up are true; otherwise, the conjunction is false. Let us gather these ideas into a single definition.
DEFINITION 1.1 If p and q are propositions, the sentence "p and q," denoted p [conjunction] q, is called the conjunction of p and q, and its truth values are given by the following table:
pqp [conjunction] q
TT T
TF F
FT F
FF F
In other words, the conjunction p [conjunction] q is true only in the case that both p and q are true; otherwise, it is false.
NOTATION RULE 1.1 If we want a variable to represent a specific proposition, we will use the following notation:
p: Some men are mathematicians.
Example 1 Consider the sentences
r: 2 + 3 = 5
s: 5 - 1 = 3
Obviously, r is true and s is false. Therefore, the second row of the table defining the conjunction tells us that the following sentence is false:
r [conjunction] s
From a different row of the table, you should be able to determine whether the following sentence is true or false by replacing s and r by p and q, respectively:
s [conjunction] r
Suppose we wish to write the denial of a proposition such as
Surgery is recommended.
We write
Surgery is not recommended.
This negation of the original sentence was easily obtained. However, the denial or negation of a proposition is not always so transparent. Consider the proposition
All ducks waddle.
Which of the following sentences would you choose as the negation of this sentence?
p1: All ducks do not waddle.
p2: Some ducks waddle.
p3: Some ducks do not waddle.
p4: Not all ducks waddle.
p5: It is not the case that all ducks waddle.
If you chose p3, p4, and p5, you are thinking clearly. In any case, the point is that some care must be taken when constructing the denial of a proposition. With these ideas in mind we can frame the following definition.
DEFINITION 1.2 If p is a proposition, the sentence "not p," denoted ~p, is called the negation of p, and its truth values are given by the following table:
p ~p
T
F
F
T
This table elucidates the nearly obvious observation that if a proposition is true, its denial is false, and vice versa. Tables such as this one and the one given in Definition 1.1 are called truth tables.
Example 2 Consider the sentences
p: Some oysters are silent.
~p: No oysters are silent.
Here the temptation is to write
Some oysters are not silent.
~p: Some oysters are not silent.
We would be mistaken, however, since it would then be possible for both p and ~p to be true.
Forming the negation of a proposition is sometimes a tricky task. Truth tables can be used to determine under what conditions the new sentence is true, and then can be used as an aid in giving alternative English-language formulations of such a sentence. Before considering an example, let us clarify some of the notation involved.
NOTATION RULE 1.2 By the notation
(~p) [conjunction] q
we wish to denote only the negation of p in the sentence p [conjunction] q. If we wish to denote the negation of p [conjunction] q (or the negation of any other sentence involving more than one variable), we will use parentheses as follows:
~(p [conjunction] q)
Example 3 This notational device exists to remove the possible ambiguity from such sentences as ~p [conjunction] q. Consider
p: Mark is intelligent.
q: Mark is wealthy.
Then we have:
(~p) [conjunction] q: Mark is not intelligent and Mark is wealthy.
and
~(p [conjunction] q): It is not the case that Mark is intelligent and wealthy.
It should be clear that these two sentences do not have the same meaning. The conditions under which ~(p [conjunction] q) is true can be inferred from the following truth table:
p q p [conjunction] q ~(p [conjunction] q)
T T
T
F
T F
F
T
F T
F
T
F F
F
T
Thus unless Mark is wealthy and intelligent, the sentence ~(p [conjunction] q) is true. This fact should be a clue to the reader who desires to translate sentences such as ~(p [conjunction] q) into English sentences less contrived than those beginning "It is not the case that ..."
As we shall soon see, such truth tables are enormously effective in the determination of the validity of certain forms or patterns of argumentation.
PROBLEM SET 1.2
In each of the following problems (1–6), you are given two propositions. The truth values of these propositions should be obvious to you. Use the truth table defining the conjunction to decide whether the conjunction of each pair is true or false. Indicate which row of the table helped you decide.
1. Some Americans are Republicans. Lansing is the capital of Michigan.
2. A triangle has three sides. Abraham Lincoln was the thirty-third president of the United States.
3. Some men are not handsome. Rudyard Kipling is the author of Gulliver's Travels.
4. 15 - 13 = 6; 21 + 5 = 26
5. College students do not study geometry. Getting an education is a waste of time.
6. Liberia is on the western coast of Africa. The United Nations headquarters is in New York City.
Try to write a sentence that is the negation of each of the following sentences (7–12) without using such crutches as "it is not the case that."
7. Mathematics is interesting.
8. There is no further need to worry.
9. Some Marxists live in California.
10. Every line is a set of points.
11. Nuts can be found in grocery stores and on nut trees.
12. A buffalo can always toss one over a gate.
13. Complete the following truth tables:
(a)
p q ~p (~p) [conjunction] q
T T
T F
F T
F F
(b)
p q ~q p [conjunction] ~(q)
T T
T F
F T
F F
14. In the manner of Example 3, and problem 13, construct the truth table for the sentence (~p) [conjunction] (~q).
1.3 PROPOSITIONS AND TRUTH TABLES CONTINUED: DISJUNCTION, IMPLICATON, AND EQUIVALENCE
In the English language, the word "or" is used in a manner that is often somewhat ambiguous, as in the following sentence:
Bertrand Russell or Alfred North Whitehead wrote the Principia Mathematica.
Here it is possible to interpret the "or" in the "either ... or ..." sense as well as the "and/or" sense. We will always assume that "or" is used in this second inclusive sense. A sentence employing "or" in this manner is called a disjunction.
DEFINITION 1.3 If p and q are propositions, the sentence "p or q," denoted p [disjunction] q, is called the disjunction of p and q, and its truth values are given by the following table
P q p [disjunction] q
T T
T
T F
T
F T
T
F F
F
As the table indicates, an "or" sentence is false whenever both the sentences making it up are false; otherwise, the disjunction is true.
Example 1 Consider the sentences
r: 2 + 3 = 5
s: 5 - 1 = 3
The second row of the table defining the disjunction tells us that the following sentence is true:
r [disjunction] s
Example 2 It should be clear that the sentence
p [disjunction] (~p)
is always true, since the definition of negation assures us that one of the component parts of this sentence is true. Thus if you ask the mathematician the question: Do you want to stay home or go out this evening? his answer may well be a truthful but uncommunicative "Yes!"
When one idea is the consequence of another, we often say that the first idea implies the second. Such a sentence is called an implication or a conditional sentence. The sentence "A falling barometer implies that a storm is on the way" is an implication.
(Continues...)
Excerpted from A First Course in Geometry by Edward T. Walsh. Copyright © 2014 Edward T. Walsh. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses.
Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter includes a problem set arranged in order of increasing difficulty as well as review exercises and annotated references suggesting sources for further study. In addition to three helpful Appendixes, the book concludes with answers and hints for selected problems.
Dover (2014) republication of the edition originally published by the Rinehart Press, San Francisco, 1974.
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