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| ISBN-13: | 9780486652528 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/01/1987 |
| Series: | Dover Books on Mathematics Series |
| Pages: | 350 |
| Product dimensions: | 5.37(w) x 8.50(h) x (d) |
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Probability: An Introduction
By Samuel Goldberg
Dover Publications, Inc.
Copyright © 1960 Samuel GoldbergAll rights reserved.
ISBN: 978-0-486-65252-8
CHAPTER 1
SETS
1. Examples of sets; basic notation
The concept of a set, whose fundamental role in mathematics was first pointed out in the work of the mathematician Georg Cantor (1845-1918), has significantly affected the structure and language of modern mathematics. In particular, the mathematical theory of probability is now most effectively formulated by using the terminology and notation of sets. For this reason, we devote Chapter 1 to the elementary mathematics of sets. Additional topics in set theory are included throughout the text, as the need for this material becomes apparent.
The notion of a set is sufficiently deep in the foundation of mathematics to defy being defined (at the level of this book) in terms of still more basic concepts. Hence, we can only aim here, by taking advantage of the reader's knowledge of the English language and his experience with the real and conceptual world, to make clear the denotation of the word "set."
A set is merely an aggregate or collection of objects of any sort: people, numbers, books, outcomes of experiments, geometrical figures, etc. Thus, we can speak of the set of all integers, or the set of all oceans, or the set of all possible sums when two dice are rolled and the number of dots on the uppermost faces are added, or the set consisting of the cities of Cambridge and Oberlin and all their residents, or the set of all straight lines (in a given plane) which pass through a given point.
The collection of objects must be well-defined, by which we mean that, for any object whatsoever, the question "Does this object belong to the collection?" has an unequivocal "yes" or "no" answer. It is not necessary that we personally have the knowledge required to decide which answer is correct. We must know only that, of the answers "yes" and "no," exactly one is correct.
Let us also agree that no object in a set is counted twice; i.e., the objects are distinct. It follows that, when listing the objects in a set, we do not repeat an object after it is once recorded. For example, according to this convention, the set of letters in the word "banana" is a set containing not six letters, but rather the three distinct letters b, a, and n.
The following definition summarizes our discussion to this point and introduces some additional terminology and notation.
Definition 1.1. A set is a well-defined collection of distinct objects. The individual objects that collectively make up a given set are called its elements, and each element belongs to or is a member of or is contained in the set. If a is an object and A a set, then we write a [member of] A as an abbreviation for "a is an element of A" and a [not member of] [not member of] for "a is not an element of A." If a set has a finite number of elements, then it is called a finite set; otherwise it is called an infinite set.
We are relying on the readers knowledge of the positive integers 1, 2, 3, ..., the so-called counting or natural numbers. This is an infinite set of numbers. To say that a set is finite means that one can enumerate the elements of the set in some order, then count these elements one by one until a last element is reached. Let us note that it is possible for a set, like the set of grains of sand on the Coney Island beach, to have a fantastically large number of elements and nevertheless be a finite set.
A set is ordinarily specified either by (i) listing all its elements and enclosing them in braces (the so-called roster method of defining the set), or by (ii) enclosing in braces a defining property and agreeing that those objects that have the property, and only those objects, are members of the set, We discuss these important ideas further and introduce additional notation in the following examples.
Example 1.1. The set whose elements are the integers 0, 5, and 12 is a finite set with three elements. If we denote this set by A, then it is conveniently written using the roster method: A = {0, 5, 12}. The statements "5 [member of] A" and "6 [not member of] A" are both true.
Example 1.2. If we write V = {a, e, i, o, u}, then we have defined the set F of vowels in the English alphabet by listing its five elements. To specify V by a defining property we write
V = {x | x is a vowel in the English alphabet},
which is read "V is the set of those elements x such that x is a vowel in the English alphabet." Braces are always used when specifying a set; the vertical bar | is read "such that" or "for which." The symbol x is of course merely a place-holder; any other symbol will do just as well. For example, we can also write
V = {* | * is a vowel in the English alphabet}.
A slight modification of this notation is often used. Let us first introduce the set A to stand for the set of all letters of the English alphabet. Then we write
V = {* [member of] A | * is a vowel},
which is read "V is the set of those elements * of A such that * is a vowel."
Example 1.3. The set B = {-2, 2} is the same set as {x [member of] R | x2 = 4}, where R is the set of all real numbers. The set {x [member of] R | x2 = -1} has no elements, since the square of any real number is nonnegative. But if C is the set of all complex numbers, then {x [member of] C | x2 = -1} contains the elements i = [square root of -1] and -i.
Example 1.4 A prime number is a positive integer greater than 1 but divisible only by 1 and itself. A proof of the fact that the set {p | p is a prime number} is an infinite set was given by Euclid (?330-275 B.C.) in the ninth book of his Elements. Strictly speaking, the roster method is unavailable for infinite sets, since it is not possible to list all the members and have explicitly before one a totality of elements making up an infinite set. The notation
{2, 3, 5, 7, 11, 13, 17, 19, ...},
in which some of the elements of the set are listed followed by three dots which take the place of et cetera and stand for obviously understood omissions of one or more elements, is an often used but logically unsatisfactory way out of this difficulty. (See Problem 1.3.) To specify an infinite set correctly, one must (as we did when we introduced the set of prime numbers) cite a defining property of the set.
Example 1.5. If a rectangular coordinate system (with x-axis and y-axis) is introduced in a plane, then each point of the plane has an x-coordinate and a y-coordinate, and can be represented, as in Figure 1(a), by an ordered pair of real numbers. In analytic geometry, one is interested in sets of points whose coordinates meet certain requirements. For example, the set {(x, y) | y = x} is the set of all points (in a plane) with equal x- and y-coordinates. This infinite set of points makes up the straight line L, a portion of which is sketched in Figure 1(a), passing through the origin O and bisecting the first and third quadrants. We say that the line L is the graph of the set {(x, y) | y = x}. Similarly, the entire x-axis is the graph of the set {(x, y) | y = 0}, and the positive x -axis is the graph of the set {(x, y) | x > 0 and y = 0}. The set {(x, y) | x > 0 and y > 0} is the set of points whose x- and y-coordinates are both positive. Thus, the graph of this set is the entire first quadrant (axes excluded), as indicated in Figure 1(b).
We see that a relation (in the form of equalities or inequalities between x and y) can be considered a set-selector, and the graph pictures the set of those points (from among all in the plane) selected by the requirement that their coordinates satisfy the given relation.
Although it may seem strange at first, it turns out to be convenient to talk about sets that have no members.
Definition 1.2. A set with no members is called an empty or null set.
The set {x [member of] R | x2 = — 1} in Example 1.3 is an empty set. Another example is obtained by considering the set of all paths by which the line drawing of a house in Figure 2 can be traced without lifting one's pencil or retracing any line segment. Whether this set is empty or not is of some interest, since to assert that it is empty is to say that the figure cannot be traced under the prescribed conditions. (Let the reader convince himself that this set is indeed empty.) As our work develops, we shall see many other less frivolous reasons for introducing the notion of an empty set.
We conclude this ground-breaking section with one more definition.
Definition 1.3. Two sets A and B are said to be equal and we Write A = B if and only if they have exactly the same elements. If one of the sets has an element not in the other, they are unequal and we write A ≠ B.
Thus A = B means that every element of A is also an element of B and every element of B is an element of A. Equal sets are identical sets, and this identity is symbolized by the equality sign.
This definition has some interesting consequences. First, it is clear that the order in which we list the elements of a set is immaterial. For example, the set {a, b, c} is equal to the set {c, a, b}, since they do indeed have exactly the same three elements.
Also, when sets are specified by defining properties, they can be equal even though the defining properties themselves are outwardly different. Thus, the set of all even prime numbers and the set of real numbers x such that x + 3 = 5 have different defining properties, yet they are equal sets, for each contains the number 2 as its only element.
Up to now, we have been careful to speak of a set having no members as an empty set. But it is clear from Definition 1.3 that any two empty sets are equal. For to be unequal it is necessary for one of the sets to contain an element not in the other, and this is impossible since neither set contains any elements. Therefore we are justified in referring to the empty set or the null set. We denote the null set by the special symbol Ø.
PROBLEMS
1.1. We list eight sets. For each set, state whether it is finite or infinite. If finite, count the number of elements in the set. Where feasible, write the set using the roster method.
(a) The set of footnotes in Section 1.
(b) The set of letters in the word "probability."
(c) The set of odd positive integers.
(d) The set of prime numbers less than one million.
(e) The set of paths by which the following figure can be traced without lifting one's pencil or retracing any line segment:
[FIGURE OMITTED]
(f) The set of those points (in a given plane) that are exactly five units from the origin O.
(g) The set of real numbers satisfying the equation x2 - 3x + 2 = 0.
(h) The set of possible outcomes of the experiment in which one card is selected from a standard deck of 52 cards.
1.2. The following paragraph was written by a student impressed with the technical vocabulary of set theory. Rewrite in more usual English prose.
Let C be the set of Mr. and Mrs. Smith's children. C was equal to Ø until March 1, 1958. C contained exactly one element from that date until March 15, 1959 when it increased its membership by two!
1.3. To illustrate the inadequacy of displaying a few elements of a set and indicating the other elements by three dots, consider the set A of all numbers of the form
n2 + (n — 1 )(n — 2 )(n — 3),
where n is any positive integer. Show that the first three elements (i.e., those obtained when n = 1, 2, 3) are 1, 4, and 9, so that one is tempted to write A = {1, 4, 9, ...}. If A is written this way on an I. Q. test, we do not hesitate to write the next element as 16. But show that the next element (obtained when n = 4) is actually 22 and not 16! Indeed, it is possible to write a defining property for a set so that its fourth element (in order of magnitude) is any number, say 94, although its first three elements are 1, 4, 9. Formulate such a defining property.
1.4. Let A = {0,1,2, 3,4}. List the elements, if any, of each of the following sets:
(a) {x [member of] A | 2x — 4 = 0}
(b) {x [member of] A | x2 — 4 = 0}
(c) {x [member of] A | x3 — 4x2 + 3x = 0}
(d) {x [member of] A | x2 = 0}
(e) {x [member of] A | x + 1 > 0}
(f) {x [member of] A | 2x + 1 ≤ 0}
(g) {x [member of] A | x2 — 5x + 4 ≥ 0}
(h) {x [member of] A | x2 — x < 0}
1.5. Let x and y be the coordinates of a point in the plane. Identify the following sets and give a geometric interpretation of your results:
(a) {(x,y) | x + y = 5 and 3x - y = 3}
(b) {(x,y) | x + y = 5 and 2x + 2y = 3}
(c) {(x,y) | x + y = 5 and 2x + 2y = 10}
1.6. Show that set equality has the following properties:
(i) Set equality is a reflexive relation; i.e., A = A for any set A.
(ii) Set equality is a symmetric relation; i.e., for any sets A and B, if A = B, then B = A.
(iii) Set equality is a transitive relation; i.e., for any sets A, B, and C, if A = B and B = C, then A = C.
(Note: A relation that is reflexive, symmetric, and transitive is called an equivalence relation,)
1.7. Determine whether A = B or A ≠ B.
(a) A = {2,4,6}, B = {4,6,2}.
(b) A = {1,2,3}, B = {Mars, Venus, Jupiter}.
(c) A = {* | * is a plane equilateral triangle}, B = {* | * is a plane equiangular triangle}.
(d) A = {x | x2 — 2x + 1 = 0}, B = {x | x — 1 = 0}.
(e) A = {x j 2s2 - 5x + 2 = 0}, B = {x | 2x* — 5x2 + 2x = 0}.
1.8. Which of the following are true? Explain.
(a) 2 = {2}, (b) 2 [member of] {2}, (c) 0 = Ø, (d) 0 [member of] Ø.
2. Subsets
Each element of the set of vowels in the alphabet is, of course, an element of the set of all letters. Similarly, each number in {2, 4, 6} is an element of the set of all even integers, and each real number in {x | x > 3} is also in {x | x > 0}. In this section, we discuss the simple but important relation between sets illustrated by these examples.
Definition 2.1. A set A is a subset of set B, denoted by A [subset or equal to] B, if each element of A is also an element of B. We agree to call the null set Ø a subset of every set.
For example, we write {1,3} [subset or equal to] {1, 2, 3}, since each of the two elements in {1, 3} belongs to {1, 2, 3}. Also, {1, 3} [subset or equal to] {x | x ≥ 1} and {1, 3} C {1, 3}. The definition of subset implies that a set is a subset of itself; i.e., A [subset or equal to] A is always true. We can express this fact using the language introduced in Problem 1.6 by saying that set inclusion (i.e., one set being a subset of another set) is a reflexive relation. It is also transitive, for if A [subset or equal to] B and B [subset or equal to] C, it follows that A [subset or equal to] C. But set inclusion is not symmetric. As a counterexample, let A = {a} and B = {a, b}. Then A [subset or equal to] B is true, but B [subset or equal to] A is false.
It is noteworthy that the definition of set equality in the preceding section was formulated in terms of the subset relation. In fact, it is merely a restatement of Definition 1.3 to say that A = B if and only if A [subset or equal to] B and B [subset or equal to] A.
Table 1 illustrates the notion of subset, and also directs our attention to a formula relating the number of subsets of a set to the number of elements in the set. We denote the number of elements in A by n(A).
From the numbers in the last column of this table, we are led to conjecture that if n is any nonnegative integer, then a set with n elements has 2n subsets. Before proving this result is true, we need to enunciate a principle that is at the heart of most counting procedures, and that is used time and again in computing probabilities.
(Continues...)
Excerpted from Probability: An Introduction by Samuel Goldberg. Copyright © 1960 Samuel Goldberg. 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.
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Table of Contents
Chapter 1 SETS1. Examples of sets; basic notation
2. Subsets
3. Operations on sets
4. The algebra of sets
5. Cartesian product sets
Chapter 2 PROBABILITY IN FINTE SAMPLE SPACES
1. Sample spaces
2. Events
3. The probability of an event
4. Some probability theorems
5. Conditional probability and compound experiments
6. Bayes' formula
7. Independent events
8. Independence of several events
9. Independent trials
10. A probability model in genetics
Chapter 3 SOPHISTICATED COUNTING
1. Counting techniques and probability problems
2. Binomial coefficients
Chapter 4 RANDOM VARIABLES
1. Random variables and probability functions
2. The mean of a random variable
3. The variance and standard deviation of a random variable
4 Joint probability functions; independent random variables
5. Mean and variance of sums of random variables; the sample mean
6. Covariance and correlation; sample mean (cont.)
Chapter 5 BINOMIAL DISTRIBUTION AND SOME APPLICATIONS
1. Bernoulli trials and the binomial distribution
2. Testing a statistical hypothesis
3. An example of decision-making under uncertainty
ANSWERS TO ODD-NUMBERED PROBLEMS
INDEX