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Lecture Notes in ActuarialMathematicsA Probability Course for theActuariesA Preparation for Exam P/1Marcel B. FinanDepartment of MathematicsArkansas Tech Universityc All Rights ReservedRevised 2020 Edition

In memory of my parentsAugust 1, 2008January 7, 2009

PrefaceThe present manuscript is a revised edition of the text that appeared firstin the year 2007. It adheres to the the SOA Syllabus October June 2020.The book is designed mainly to help students prepare for the ProbabilityExam (known as Exam P/1), the first actuarial examination administeredby the Society of Actuaries. This examination tests a student’s knowledgeof the fundamental probability tools for quantitatively assessing risk. Athorough command of calculus is assumed. However, the text will includethe mathematical background needed throughout the book.Users are encouraged to access the online site of the Society of Actuarieswww.soa.org for up-to-date information about the exam.The present version includes in many cases in depth discussions in terms ofproofs which the reader can omit. For this reason, the book is suitable for aone or two- semester course in undergraduate probability theory.Users of the book who are preparing for the exam are strongly encouragedto work out all the problems in the book. Make sure you can come out withthe solutions on your own without any outside reference. This will enhanceyour chances in performing well on the exam.The book covers all the sample problems from previous exams made availableby SOA. Such problems will be indicated by the symbol ‡.Answer keys to text problems are found at the end of the book. Mock examsare included as well. Attempt these exams after finishing all the chaptersof the book. Take these exams under the same circumstances as the actualexam.The present version of the book is made possible by a grant support fromArkansas Tech University.i

iiPREFACEFinally, this manuscript can be used for personal use or class use, but notfor commercial purposes. If you find any errors, I would appreciate hearingfrom you: [email protected] the best.Marcel B. FinanRussellville, ARAugust, 2020

ContentsPrefacei1 Set Theory Prerequisite51.1 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . 61.2 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Counting and Combinatorics2.1 The Fundamental Principle of Counting . . . . . . . . . . . .2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Review of Calculus3.1 Limits and Continuity . . . . . . .3.2 Series . . . . . . . . . . . . . . . . .3.3 The Derivative of a Function . . . .3.4 The Definite Integral . . . . . . . .3.5 Improper Integrals . . . . . . . . .3.6 Graphing Systems of Inequalities in3.7 Iterated Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Two Variables. . . . . . . . .4 Probability: Definitions and Properties4.1 Basic Definitions and Axioms of Probability . . . . . . . . .4.2 Probability of Intersection and Union . . . . . . . . . . . . .4.3 Probability and Counting Techniques . . . . . . . . . . . . .31323945535466738395106112121. 122. 132. 1405 Conditional Probability and Independence1475.1 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . 1485.2 Bayes’ Formula and the Law of Total Probability . . . . . . . 1561

2CONTENTS5.35.4Independent Events . . . . . . . . . . . . . . . . . . . . . . . . 166Odds Versus Probability . . . . . . . . . . . . . . . . . . . . . 1766 Discrete Random Variables1816.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1826.2 Probability Mass Function and Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.3 Expected Value of a Discrete Random Variable . . . . . . . . 1996.4 Expected Value of a Function of a Discrete Random Variable . 2086.5 Variance and Standard Deviation of a Discrete Random Variable2177 Commonly Used Discrete Random Variables2257.1 Uniform Discrete Random Variable . . . . . . . . . . . . . . . 2267.2 Binomial Random Variable . . . . . . . . . . . . . . . . . . . . 2327.3 The Expected Value and Variance of the Binomial Distribution 2417.4 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . 2507.5 Poisson Approximation to the Binomial Distribution . . . . . 2597.6 Geometric Random Variable . . . . . . . . . . . . . . . . . . . 2637.7 Negative Binomial Random Variable . . . . . . . . . . . . . . 2717.8 Hyper-geometric Random Variable . . . . . . . . . . . . . . . 2788 Cumulative and Survival Distribution Functions2878.1 The Cumulative Distribution Function . . . . . . . . . . . . . 2888.2 The Survival Distribution Function . . . . . . . . . . . . . . . 3039 Continuous Random Variables3099.1 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . 3109.2 The Expected Value of a Continuous Random Variable . . . . 3249.3 The Variance of a Continuous Random Variable . . . . . . . . 3339.4 Median, Mode, and Percentiles . . . . . . . . . . . . . . . . . 3399.5 The Continuous Uniform Distribution Function . . . . . . . . 3479.6 Normal Random Variables . . . . . . . . . . . . . . . . . . . . 3559.7 The Normal Approximation to the Binomial Distribution . . . 3669.8 Exponential Random Variables . . . . . . . . . . . . . . . . . 3719.9 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . 3829.10 The Distribution of a Function of a Continuous Random Variable389

CONTENTS310 Joint Distributions39710.1 Discrete Jointly Distributed Random Variables . . . . . . . . . 39810.2 Jointly Continuous Distributed Random Variables . . . . . . . 40710.3 Independent Random Variables . . . . . . . . . . . . . . . . . 41510.4 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 42510.5 Sum of Two Independent Random Variables: Discrete Case . . 42910.6 Sum of Two Independent Random Variables: Continuous Case 43610.7 Conditional Distributions: Discrete Case . . . . . . . . . . . . 44610.8 Conditional Distributions: Continuous Case . . . . . . . . . . 45610.9 Joint Probability Distributions of Functions of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46611 Properties of Expectation11.1 Expected Value of a Function of Two Random11.2 Covariance and Variance of Sums . . . . . . .11.3 The Coefficient of Correlation . . . . . . . . .11.4 Conditional Expectation . . . . . . . . . . . .11.5 Double Expectation . . . . . . . . . . . . . . .11.6 Conditional Variance . . . . . . . . . . . . . .Variables. . . . . . . . . . . . . . . . . . . . . . . . . .471. 472. 484. 492. 499. 508. 51812 Moment Generating Functions and the Central Limit Theorem52712.1 Moment Generating Functions . . . . . . . . . . . . . . . . . . 52812.2 Moment Generating Functions of Sums of Independent RVs . 53512.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . 544Sample Exam 1553Sample Exam 2567Sample Exam 3583Sample Exam 4597Sample Exam 5611Sample Exam 6625Sample Exam 7639

4CONTENTSSample Exam 8653Sample Exam 9667Sample Exam 10681Answer Keys697Bibliography785Index787

Chapter 1Set Theory PrerequisiteTwo approaches of the concept of probability will be introduced later in thebook: The classical probability and the experimental probability. The formerapproach is developed using the foundation of set theory, and a quick reviewof the theory is in order. Readers familiar with the basics of set theory suchas set builder notation, Venn diagrams, and the basic operations on sets,(unions, intersections, and complements) can skip this chapter.Set is the most basic term in mathematics. Some synonyms of a set areclass or collection. In this chapter, we introduce the concept of a set and itsvarious operations and then study the properties of these operations.Throughout this book, we assume that the reader is familiar with the following number systems and the algebraic operations and properties of suchsystems: The set of all positive integersN {1, 2, 3, · · · }. the set of whole NumbersW {0, 1, 2, 3, · · · }. The set of all integersZ {· · · , 3, 2, 1, 0, 1, 2, 3, · · · }. The set of all rational numbersaQ { : a, b Z with b 6 0}.b The set R of all real numbers.5

61.1CHAPTER 1. SET THEORY PREREQUISITESome Basic DefinitionsWe define a set as a collection of well-defined objects (called elements ormembers) such that for any given object one can assert without disputethat either the object is in the set or not but not both. Sets are usually willbe represented by upper case letters. When an object x belongs to a set A,we write x A, otherwise, we use the notation x 6 A. Also, we mention herethat the members of a set can be sets themselves.Example 1.1.1Which of the following is a set.(a) The collection of good movies.(b) The collection of men 65 years of age in a certain city.Solution.(a) Answering a question about whether a movie is good or not may be subject to dispute, the collection of good movies is not a well-defined set.(b) This collection is a well-defined set since a man is either 65 years old ornotNext, we introduce a couple of set representations. The first one is to list,without repetition, the elements of the set. For example, if A is the solutionset to the equation x2 4 0 then A { 2, 2}. The order of how elementsof a set appear is irrelevant. We refer to this type of representation as thetabular formThe other way to represent a set is to describe a property that characterizesthe elements of the set. This is known as the set-builder representation ofa set. For example, the set A above can be written as A {x x is an integersatisfying x2 4 0}. The symbol stands for the statement ”such as ”‘.We define the empty set, denoted by , to be the set with no elements. Aset which is not empty is called a non-empty set.Example 1.1.2List the elements of the following sets.(a) A {x x is a real number such that x2 1}.(b) B {x x is an integer such that x2 3 0}.Solution.(a) The real solutions to the equation x2 1 are 1. Thus, A { 1, 1}.

1.1. SOME BASIC DEFINITIONS7 (b) Since the only solutions to the given equation are 3 and 3 and bothare not integers, the set in question is the empty set. That is, B Example 1.1.3Use a property characterizing the members of the following sets.(a) A {a, e, i, o, u}.(b) B {1, 3, 5, 7, 9}.Solution.(a) A {x x is a vowel of the English alphabet}.(b) B {n n N is odd and less than 10 }The first arithmetic operation involving sets that we consider is the equalityof two sets. Two sets A and B are said to be equal if and only if they containthe same elements. We write A B. For non-equal sets we write A 6 B. Inthis case, there is at least one element in one set which is not in the otherset.Example 1.1.4Determine whether each of the following pairs of sets are equal.(a) {1, 3, 5} and {5, 3, 1}.(b) {{1}} and {1, {1}}.Solution.(a) Since the order of listing elements in a set is irrelevant, {1, 3, 5} {5, 3, 1}.(b) Since one of the sets has exactly one member and the other has two,{{1}} 6 {1, {1}}In set theory, the number of elements in a set has a special name. It iscalled the cardinality of the set. We write #(A) to denote the cardinalityof the set A. If A has a finite cardinality we say that A is a finite set. Otherwise, it is called infinite. For example, N is an infinite set.Can two infinite sets have the same cardinality? The answer is yes. If A andB are two sets (finite or infinite) and there is a bijection from A to B( i.e.,a one-to-one1 and onto2 function) then the two sets are said to have the1A function f : A 7 B is a one-to-one function if f (m) f (n) implies m n,where m, n A.2A function f : A 7 B is an onto function if for every b B, there is an a A suchthat b f (a).

8CHAPTER 1. SET THEORY PREREQUISITEsame cardinality and we write #(A) #(B).If #(A) is either finite or has the same cardinality as N then we say that Ais countable. A set that is not countable is said to be uncountable.Example 1.1.5What is the cardinality of each of the following sets?(a) .(b) { }.(c) A {a, {a}, {a, {a}}}.Solution.(a) #( ) 0.(b) This is a set consisting of one element . Thus, #({ }) 1.(c) #(A) 3Example 1.1.6(a) Show that the set A {a1 , a2 . · · · , an , · · · }, where the a0i s are distinct, iscountable.(b) Let A be the set of all infinite sequences of the digits 0 and 1. Show thatA is uncountable.Solution.(a) We first show that the map f : N 7 A defined by f (n) an is one-toone. Indeed, if f (n) f (m) then an am and this implies n m. Now, anymember in A is of the form an f (n) for some n N. Thus, A is onto. Itfollows that A is countable.(b) We will argue by contradiction3 . Suppose that A is countable with elements a1 , a2 , · · · , where each ai is an infinite sequence of the digits 0 and1. Let a be the infinite sequence with the first digit of 0 or 1 different fromthe first digit of a1 , the second digit of 0 or 1 different from the second digitof a2 , · · · , the nth digit is different from the nth digit of an , etc. Thus, a isan infinite sequence of the digits 0 and 1 which is not in A, a contradiction.Hence, A is uncountable3In logic, a proposition is a statement that is either true or false. In the method of proofby contradiction one assumes that a proposition is false and ran into a contradiction asone proceeds with the proof making the assumption that the original proposition is falseis impossible. Thus, it must be true.

1.1. SOME BASIC DEFINITIONS9Now, let A and B be two sets. We say that A is a subset of B or B asuperset of A, denoted by A B, if and only if every element of A is alsoan element of B. If there exists an element of A which is not in B, then wewrite A 6 B. A universal set U is the collection of all objects in a particularcontext or theory. All other sets in that framework constitute subsets of theuniversal set.For any set A we have A A. That is, every set has at least two subsets.Also, keep in mind that the empty set is a subset of any set.Example 1.1.7Suppose that A {2, 4, 6}, B {2, 6}, and C {4, 6}. Determine which ofthese sets are subsets of which other of these sets.Solution.Clearly, B A and C ASubsets of a universal set can be represented by circles or closed curves witha rectangle that represents the universal set. We refer to such pictorial representation as a Venn diagram.Example 1.1.8Represent A B C using Venn diagram.Solution.The Venn diagram is given in Figure 1.1.1Figure 1.1.1Next, let A and B be two sets. We say that A is a proper subset of B,denoted by A B, if A B and A 6 B. Thus, to show that A is a propersubset of B, we must show that every element of A is an element of B andthere is an element of B which is not in A.

10CHAPTER 1. SET THEORY PREREQUISITEExample 1.1.9Order the sets of numbers: W, Z, R, Q, N using .Solution.We have, N W Z Q RExample 1.1.10Determine whether each of the following statements is true or false.(a) x {x} (b) {x} {x} (c) {x} {x}(d) {x} {{x}} (e) {x} (f) {x}.Solution.(a) True (b) True (c) False since {x} is a set consisting of a single element xand so {x} is not a member of this set (d) True (e) True (f) False since {x}does not have as a listed memberNow, the collection of all subsets of a set A is of importance. We denotethis set by P(A) and we call it the power set of A.Example 1.1.11Find the power set of A {a, b, c}.Solution.P(A) { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}We have already introduced a direct method of proof known as the proofby contradiction. We next introduce another method of proof known as theproof by mathematical induction: We want to prove that some statementP (n) is true for any non-negative integer n n0 . The steps of mathematicalinduction are as follows:(i) Basis of induction: Show that P (n0 ) is true.(ii) Induction hypothesis: Assume P (n0 ), P (n0 1), · · · , P (n) are true.(iii) Induction step: Show that P (n 1) is true.Example 1.1.12(a) Use induction to show that if #(A) n then #(P(A)) 2n , wheren W.(b) If P(A) has 256 elements, how many elements are there in A?

1.1. SOME BASIC DEFINITIONS11Solution.(a) We apply induction to prove the claim. If n 0 then A and in thiscase P(A) { }. Thus, #(P(A)) 1 20 . As induction hypothesis, suppose that if #(A) k, where k 0, 1, 2, · · · , n, then #(P(A)) 2k . Let B {a1 , a2 , · · · , an , an 1 }. Then P(B) consists of all subsets of {a1 , a2 , · · · , an }together with all subsets of {a1 , a2 , · · · , an } with the element an 1 added tothem. Hence, #(P(B)) 2n 2n 2 · 2n 2n 1 .(b) Since #(P(A)) 256 28 , by (a) we have #(A) 8Example 1.1.13Use induction to show thatnX(2i 1) n2 , n N.i 1Solution.2If n 1 we have 1 2(1) 1 1Xi 1(2i 1). Suppose that the resultis true for k 1, 2, · · · , n. We will show that it is true for n 1. Indeed,n 1nXX(2i 1) (2i 1) 2(n 1) 1 n2 2n 2 1 (n 1)2i 1i 1

12CHAPTER 1. SET THEORY PREREQUISITEPractice ProblemsProblem 1.1.1Consider the experiment of rolling a die. List the elements of the set A {x xshows a face with prime number}. Recall that a prime number is a positiveinteger greater than 1 with only two different divisors: 1 and the numberitself.Problem 1.1.2Consider the random experiment of tossing a coin three times.(a) Let S be the collection of all outcomes of this experiment. List the elements of S. Use H for head and T for tail.(b) Let E be the subset of S with more than one tail. List the elements ofE.(c) Suppose F {T HH, HT H, HHT, HHH}. Write F in set-builder notation.Problem 1.1.3Consider the experiment of tossing a coin three times. Let E be the collectionof outcomes with at least one head and F the collection of outcomes of morethan one head. Compare the two sets E and F.Problem 1.1.4Recall that a standard deck of 52 playing cards can be described as follows:hearts (red)clubs (black)diamonds (red)spades (black)AceAceAceAce2 3 4 5 6 7 8 9 10 Jack Queen King2 3 4 5 6 7 8 9 10 Jack Queen King2 3 4 5 6 7 8 9 10 Jack Queen King2 3 4 5 6 7 8 9 10 Jack Queen KingCards labeled Ace, Jack, Queen, or King are called face cards.A hand of 5 cards is dealt from a deck of 52 cards. Let E be the event thatthe hand contains 5 aces. List the elements of E.Problem 1.1.5Prove the following properties:(a) Reflexive Property: A A.(b) Antisymmetric Property: If A B and B A then A B.(c) Transitive Property: If A B and B C then A C.

1.1. SOME BASIC DEFINITIONS13Problem 1.1.6Prove by using mathematical induction that1 2 3 ··· n n(n 1), n N.2Problem 1.1.7Prove by using mathematical induction that12 22 32 · · · n2 n(n 1)(2n 1), n N.6Problem 1.1.8Use induction to show that (1 x)n 1 nx for all n W, where x 1.Problem 1.1.9Use induction to show that1 a a2 · · · an 1 1 an, a 6 1, n N.1 aProblem 1.1.10Subway prepared 60 4-inch sandwiches for a birthday party. Among thesesandwiches, 45 of them had tomatoes, 30 had both tomatoes and onions,and 5 had neither tomatoes nor onions. Using a Venn diagram, how manysandwiches did he make with(a) tomatoes or onions?(b) onions?(c) onions but not tomatoes?Problem 1.1.11A camp of international students has 110 students. Among these eakspeakEnglish,Spanish,French,English and Spanish,English and French,Spanish and French,all three languages.

14CHAPTER 1. SET THEORY PREREQUISITEHow many students speak(a) English and Spanish, but not French,(b) neither English, Spanish, nor French,(c) French, but neither English nor Spanish,(d) English, but not Spanish,(e) only one of the three languages,(f) exactly two of the three languages.Problem 1.1.12An experiment consists of the following two stages:(1) a fair coin is tossed(2) if the coin shows a head, then a fair die is rolled; otherwise, the coin isflipped again.An outcome of this experiment is a pair of the form (outcome from stage1, outcome from stage 2). Let S be the collection of all outcomes. List theelements of S and then find the cardinality of S.Problem 1.1.13Show that the function f : R 7 R defined by f (x) 3x 5 is one-to-oneand onto.Problem 1.1.14Find #(A) if #(P(A)) 32.Problem 1.1.15Consider the function f : N 7 Z defined by n,if n is even2f (n) n 1,if n is odd.2(a) Show that f (n) f (m) only if n and m have the same parity, i.e., eitherboth are even or both are odd.(b) Show that Z is countable.Problem 1.1.16Let A be a non-empty set and f : A 7 P(A) be any function. Let B {a A a 6 f (a)}. Clearly, B P(A). Show that there is no b A such thatf (b) B. Hence, there is no onto map from A to P(A).

1.1. SOME BASIC DEFINITIONS15Problem 1.1.17Use the previous problem to show that P(N) is uncountable.Problem 1.1.18Show that the sets (0, ) and R have the same cardinality.Problem 1.1.19Show that the function f : N N 7 N defined by f (n, m) 2n 3m isone-to-one.Problem 1.1.20A marketing survey indicates that 60% of the population owns a laptop,30% owns a desktop computer, and 20% owns both a laptop and a desktopcomputers. What percent of the population owns a laptop or a desktop, butnot both?

161.2CHAPTER 1. SET THEORY PREREQUISITESet OperationsIn this section we introduce various operations on sets and study the properties of these operations.ComplementsLet U be a universal set and A, B be two subsets of U. The absolute complement of A (See Figure 1.2.1(I)) is the setAc {x U x 6 A}.Example 1.2.1Find the absolute complement of A {1, 2, 3} if U {1, 2, 3, 4, 5, 6}.Solution.From the definition, Ac {4, 5, 6}The relative complement of A with respect to B (See Figure 1.2.1(II))is the setB A {x U x B and x 6 A}.Figure 1.2.1Example 1.2.2Let A {1, 2, 3} and B {{1, 2}, 3}. Find A B.Solution.The elements of A that are not in B are 1 and 2. Thus, A B {1, 2}Union and IntersectionIn the remaining of this book, all sets are assumed to be subsets of a universalset. Given two sets A and B. The union of A and B is the setA B {x x A or x B}

1.2. SET OPERATIONS17where the ‘or’ is inclusive.(See Figure 1.2.2(a))Figure 1.2.2The above definition can be extended to more than two sets. More precisely,if A1 , A2 , · · · , are sets then [An {x x Ai f or some i N}.n 1The intersection of A and B is the set (See Figure 1.2.2(b))A B {x x A and x B}.If A B we say that A and B are disjoint sets.Example 1.2.3Express each of the following sets in terms of the sets A, B, and C as well asthe operations of absolute complement, union and intersection. In each casedraw the corresponding Venn diagram.(a) x belongs to at least one of the sets A, B, C;(b) x belongs to at most one of the sets A, B, C;(c) x is none of the sets A, B, C;(d) x belongs to all three sets A, B, C;(e) x belongs to exactly one of the sets A, B, C ;(f) x belongs to A and B but not C.Solution.The Venn diagrams are shown in Figure 1.2.3.(a) A B C(b) (A B c C c ) (Ac B C c ) (Ac B c C) (Ac B c C c )(c) (A B C)c Ac B c C c(d) A B C

18CHAPTER 1. SET THEORY PREREQUISITE(e) (A B c C c ) (Ac B C c ) (Ac B c C)(f) A B C cFigure 1.2.3Example 1.2.4Find a simpler expression of [(A B) (A C) (B c C c )] assuming allthree sets A, B, C intersect.Solution.Using the Venn diagram in Figure 1.2.4, one can easily see that [(A B) (A C) (B c C c )] A [A (B C)] A B CFigure 1.2.4Example 1.2.5Let A and B be two non-empty sets. Write A as the union of two disjointsets.

1.2. SET OPERATIONS19Solution.Using a Venn diagram one can easily see that A B and A B c are disjointsets such that A (A B) (A B c )The concept of intersection can be extended to a countable number of sets.Given the sets A1 , A2 , · · · , we define \An {x x Ai f or all i N}.n 1Example 1.2.6For each positive integer n, we define An {n}. Find \An .n 1Solution. \An Clearly,n 1Theorem 1.2.1Let A, B and C be subsets of U. We have(a) Commutative law: A B B A and A B B A.(b) Associative law: A (B C) (A B) C and (A B) C A (B C).Proof.(a) We have: x A B x A and x B x B and x A x B A. Similar proof holds for the union where the “and” is replacedby “or”.(b) We have: x A (B C) x A and x B C x A and (x B and x C) (x A and x B) and x C x (A B) C. Similarproof holds for the unionThe following theorem establishes the distributive laws of sets.Theorem 1.2.2If A, B, and C are subsets of U then(a) A (B C) (A B) (A C).(b) A (B C) (A B) (A C).

20CHAPTER 1. SET THEORY PREREQUISITEProof.See Problem 1.2.15The following theorem presents the relationships between (A B)c , (A B)c , Ac and B c .Theorem 1.2.3 (De Morgan’s Laws)Let A and B be subsets of U. We have(a) (A B)c Ac B c .(b) (A B)c Ac B c .Proof.We prove part (a) leaving part(b) as an exercise for the reader.(a) Let x (A B)c . Then x U and x 6 A B. Hence, x U and (x 6 Aand x 6 B). This implies that (x U and x 6 A) and (x U and x 6 B).It follows that x Ac B c .Conversely, let x Ac B c . Then x Ac and x B c . Hence, x 6 A andx 6 B which implies that x 6 (A B). Hence, x (A B)cRemark 1.2.1Using mathematical induction, De Morgan’s laws are valid for any countablenumber of sets. That is!c [\An Acnn 1and \n 1!cAnn 1 [Acn .n 1Example 1.2.7An assisted living agency advertises its program through videos and booklets.Let U be the set of people solicited for the agency program. All participantswere given a chance to watch a video and to read a booklet describing theprogram. Let V be the set of people who watched the video, B the set ofpeople who read the booklet, and C the set of people who decided to enrollin the program.(a) Describe with set notation: “The set of people who did not see the videoor read the booklet but who still enrolled in the program”(b) Rewrite your answer using De Morgan’s law and and then restate theabove.

1.2. SET OPERATIONS21Solution.(a) (V B)c C.(b) (V B)c C V c B c C the set of people who did not watch thevideo, did not read the booklet, but did enrollIf Ai Aj for all i 6 j then we say that the sets in the collection{An } n 1 are pairwise disjoint.Example 1.2.8Find three sets A, B, and C that are not pairwise disjoint but A B C .Solution.One example is A B {1} and C Example 1.2.9Throw a pair of fair dice. Let A be the event the total is 5, B the event thetotal is even, and C the event the total is divisible by 9. Show that A, B,and C are pairwise disjoint.Solution.We haveA {(1, 4), (2, 3), (3, 2), (4, 1)}B {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6)(3, 1), (3, 3), (3, 5), (4, 2),(4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}C {(3, 6), (4, 5), (5, 4), (6, 3)}.Clearly, A B A C B C Next, we establish the following rule of counting.Theorem 1.2.4 (Inclusion-Exclusion Principle)Suppose A and B are finite sets. Then(a) #(A B) #(A) #(B) #(A B).(b) If A B , then #(A B) #(A) #(B).(c) If A B, then #(A) #(B).

22CHAPTER 1. SET THEORY PREREQUISITEProof.(a) Indeed, #(A) gives the number of elements in A including those thatare common to A and B. The same holds for #(B). Hence, #(A) #(B)includes twice the number of common elements. Therefore, to get an accuratecount of the elements of A B, it is necessary to subtract #(A B) from#(A) #(B). This establishes the result.(b) If A and B are disjoint then #(A B) 0 and by (a) we have #(A B) #(A) #(B).(c) If A is a subset of B then the number of elements of A cannot exceed thenumber of elements of B. That is, #(A) #(B)Example 1.2.10The State Department interviewed 35 candidates for a diplomatic post inAlgeria; 25 speak arabic, 28 speak french, and 2 speak neither languages.How many speak both languages?Solution.Let F be the group of applicants that speak french, A those who speak arabic.Then F A consists of those who speak both languages. By the InclusionExclusion Principle, we have #(F A) #(F ) #(A) #(F A). Thatis, 33 28 25 #(F A). Solving for #(F A) we find #(F A) 20Cartesian ProductThe notation (a, b) is known as an ordered pair of elements and is definedby (a, b) {{a}, {a, b}}.The Cartesian product of two sets A and B is the setA B {(a, b) a A, b B}.The idea can be extended to products of any number of sets. Given n setsA1 , A2 , · · · , An the Cartesian product of these sets is the setA1 A2 · · · An {(a1 , a2 , · · · , an ) : a1 A1 , a2 A2 , · · · , an An }where, we define(a1 , a2 , · · · , an ) ((a1 , a2 , · · · , an 1 ), an ), n 2.Example 1.2.11Show that (a1 , a2 , · · · , an ) (b1 , b2 , · · · , bn ) if and only if ai bi for i 1, 2, · · · , n.

1.2. SET OPERATIONS23Solution.The proof is by induction on n 2. For the basis of induction, we have(a1 , a2 ) (b1 , b2 ) if and only if {{a1 }, {a1 , a2 }} {{b1 }, {b1 , b2 }} and thisis equivalent to {a1 } {b1 } (i.e., a1 b1 ) and {a1 , a2 } {b1 , b2 }. Thus,a1 b1 and a2 b2 . For the induction hypothesis, suppose that the result holds for k 2, 3, · · · , n. For the induction step, we must show that(a1 , a2 , · · · , an 1 ) (b1 , b2 , · · · , bn 1 ) is equivalent to ai bi for i 1, 2, · · · , n 1. Indeed,(a1 , a2 , · · · , an 1 ) (b1 , b2 , · · · , bn 1 ) ((a1 , a2 , · · · , an ), an 1 ) ((b1 , b2 , · · · , bn ), bn 1 ) (a1 , a2 , · · · , an ) (b1 , b2 , · · · , bn ), an 1 bn 1 ai bi , i 1, 2, · · · , n 1Example 1.2.12Consider the experiment of tossing a fair coin n times. Represent the samplespace as a

Exam (known as Exam P/1), the rst actuarial examination administered . Sample Exam 6625 Sample Exam 7639. 4 CONTENTS Sample Exam 8653 Sample Exam 9667 Sample Exam 10681 Answer Keys697 Bibliography785 Index787. Chapter 1 Set Theory Prerequisite Two approaches of t