1. The collection of the vowels in the word probability. 2. The collection of real numbers that satisfy the equation x 9 = 0.

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1 C HPTER 1 SETS I. DEFINITION OF SET We begin our study of probability with the disussion of the basi onept of set. We assume that there is a ommon understanding of what is meant by the notion of a olletion of things, a set of objets, or a group of artiles. Therefore we will not attempt to define this fundamental idea of a olletion of things, but we will use it to give a definition of the mathematial usage of the term set. What is a set? set is a well-defined olletion of distint objets. The objets in the set are alled the elements or members of the set. Capital letters,,c, are usually used to denote sets and lowerase letters a,b,, to denote the elements of a set. s we shall see by some examples, it is not enough to say that a set is simply a olletion of objets. The distinguishing idea is that the olletion must be welldefined. That is, for a olletion S to be set it must always be possible to determine whether or not some given objet x is an element of S. Examples 1.1: 1. The olletion of the vowels in the word probability.. The olletion of real numbers that satisfy the equation x 9 = The olletion of two-digit positive integers that are divisible by The olletion of great football players in the National Football League. 5. The olletion of intelligent members of the nited States Congress. The first three olletions are sets beause in eah ase it is ertainly possible to determine whether or not some given objet x is an element of the set. On the other hand, olletions 4 and 5 are not sets; for a given NFL football player x some people might onsider him to be great, others might not. Similarly, different people would probably ome up with different olletions of intelligent members of Congress. Colletion 5 also suggests the need for a speial set, namely the set that has no elements. The set with no elements is alled the empty set or the null set. The empty 1

2 set is denoted by the symbol Ø. For a onrete example, let be the set of real numbers x that satisfy the equation x + 1 = 0, whih is the same as x = 1. Sine the square of any real number is always a nonnegative real number, there are no real numbers whih satisfy this equation. Thus is the empty set ( is Ø). Other examples whih justify the need for the empty set will be given below. Speifying a set. There are various ways to denote a set. When the number of the elements in the set is small, or when the elements have an obvious pattern, you an simply list the elements, enlosed within braes. This method is alled the roster method. Examples 1.: 1. The set of vowels in the word probability is the set C { a, o, i} =. Note that the letter i appears twie in the word probability but we list it one - a set is a olletion of distint elements.. The set of real numbers that satisfy the equation x 9 = 0 is the set S = 3,3. { } 3. The set of two-digit positive integers that are divisible by 5 is the set K = { 10,15,0,5,...,95}. The three dots (alled ellipsis) indiate that the list of elements ontinues in the same pattern established by the first few elements. In eah of these examples it is possible to ount the number of the elements in the set; C has three elements, S has two elements, K has eighteen elements (verify this). These are examples of finite sets. On the other hand, the set of odd positive integers, whih an be denoted by P = { 1,3,5,7,... } has infinitely many elements and is an example of an infinite set. The set of integers, the set of rational numbers, and the set of real numbers are other examples of infinite sets. The number of elements in a given set is alled the ardinality of and is denoted by n (). y definition, n ( Ø) = 0 (the empty set has no elements). If Ø, then n () is either a positive integer ( is a finite set), or n () = ( is an infinite set). You should be able to determine whether a given set is finite or infinite. Our primary interest in this material is in finite sets, but examples involving infinite sets will ome up from time to time. Methods for ounting the number of elements in a finite set will be onsidered in the next setion. There are situations where a set has a large number of elements whih an not be listed easily, or at all, or where the elements are determined by some harateristi property. For example, it is not possible to list all the elements in the set Q of rational numbers and, although possible, it would be unwieldy to list the elements of

3 the set T of states in the nited States. So we need an alternative to the roster method to denote a set. Set builder notation is used to desribe a set whose elements have a defining harateristi property: { x x has a speifi property P} =, read the set of all x suh that x has property P. Examples 1.3: 1. The set of rational numbers Q an be denoted by Q = x x is a rational number or by { } { x x = p / q where p and q are integers and 0} Q = q. The set T of states in the nited States is denoted by T = x x is a state in the nited States. { } niversal Set. Conneted with a given set there is almost always a surrounding ontext, a set that ontains all the objets of interest. For example, in onsidering the set C of vowels in the word probability, the ontext is the letters of the English alphabet. We do not have to onern ourselves as to whether the number 3 belongs to C or whether a ertain person is in C; learly they are not. We only have to worry about whether or not a given letter is in C. Similarly, an enompassing ontext for the set P of the odd positive integers is the set N of positive integers; we do not have to be onerned as to whether or not the letter a is in P, learly it is not. set that inludes all of the elements under onsideration in a partiular disussion is alled a universal set. The purpose of the universal set is to fous attention on the objets of interest. The need for it will beome apparent in our disussion of set operations below. However, as shown in the following examples, there may be more than one universal set assoiated with a given set. Examples 1.4: 1. C { a, o, i} ontext is the letters of the alphabet, the set { a, b,,..., x, y, z} for C. However, if we are onerned only with vowels, then = { a, e, i, o, u} = is the set of vowels in the word probability. Sine the overall = is a universal set would also serve as a universal set for C.. Suppose we roll a standard, six-sided die and ask whether the result is a prime number. Then we are asking whether the outome is an element of the set 3

4 D = {, 3, 5} that an result from rolling the die, namely = { 1,, 3, 4, 5, 6} the elements of D are positive integers, so we ould use the set = { 1,, 3,... }. n obvious universal set for D is the set of all possible outomes. On the other hand, N as a universal set for D; the elements of D are also real numbers so we ould use the set R = {x x is a real number} as a universal set for D. The set an be thought of as the minimal universal set for D. II. SET OPERTIONS Let be a set and let x be some objet. We know that x either is, or is not, an element of. The statement x is an element of (also stated x is a member of or x is in ) is denoted by x ; the statement x is not an element of is denoted by x. Examples.1: 1. Let V be the set of vowels: V { a, e, i, o, u} =. Then e V and b V.. Let P be the set of prime numbers: P = {, 3, 5, 7,... } set.) Then 17 P and 39 P.. (Note: P is an infinite Equality. Given two sets and. Our first onern is the question of equality; are and the same set, or are they different sets. Two sets and are equal, denoted in the familiar way =, if and only if they have exatly the same elements. If and do not have exatly the same elements then is not equal to, denoted by. Examples.: Let = { a, b,, d}, = { a, b,, d, e, f }, C { b, d, a, } =. Then 1. = C ( and C have exatly the same elements, the order in whih the elements of a set are listed is immaterial). ( and do not have exatly the same elements; e, f, but e, f ). Subset. Equality is not the only relationship between two sets. The sets and in the example above are not equal beause they do not have exatly the same elements. However, note that eah element of is also and element of ; the set is inside. If and are sets and eah element of is also an element of, then is said to be a subset of, symbolized by. 4

5 Example.3: Let = {1,,3,4,5,6 }, = {,3,5}, C = {1,,3,57}. Determine whih of the following statements are true. (a) 5 e (b) 5 Œ () {5} Œ (d) {5} œ (e) Œ (f) C Œ (g) Œ Solution: (a) True; 5 is an element of. (b) False; 5 annot be a subset of sine 5 is not a set. () True; {5} is a set and every element of {5} is an element of. (d) False; if {5} œ, then the set {5} must be one of the elements of ; the element 5 is in, the set {5} is not in. (e) True; every element in is also in. (f) False; the element 7 is in C but it is not in. (g) True; every element in is an element in. s illustrated in the example above, a set is a subset of itself; for any set,. What about the empty set, Ø? The onvention is that the empty set is a subset of every set. One way to justify this is by asking: re there any elements in Ø whih are not in a given set? Sine there are no elements in Ø, the answer is no. Thus, Ø. set is said to be a proper subset of if is a subset of and, Ø. Example.4: Let = {a, b, }. The proper subsets of are: {a}, {b}, {}, {a, b}, {a, }, {b, }. nion. Given two sets and.the union of and, denoted by elements that are either in or in. In symbols, { x x x } = or., is the set of 5

6 The use of the word or here is inlusive; that is x is either in or in means either x is in, or x is in, or x is in both and. Intersetion. Given two sets and. The intersetion of and, denoted by of elements that are in both and. In symbols, Complement. = { x x x } and., is the set The last set operation we will define is the omplement of a set. Here we will have to use the notion of a universal set introdued above. Let be a set and be a universal set for. The omplement of, denoted by, is the set of elements in that are not in. That is, = { x x } Examples.5: Let (universal set) = {1,, 3, 4,, 0}. Let, and C be subsets of with = {1, 3, 5,, 19}, = {x x is even and 4 < x <1}, and C = {x x is an odd prime number}. Find, in roster form, eah of the following sets: (a) (b) () (d) C (e) C Solution: = {6, 8, 10} and C = {3, 5, 7, 11, 13, 17, 19}. (a) = {1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19}. (b) C ; C = {3, 5, 7, 11, 13, 17, 19} = C. () C = {1,, 4, 6, 8, 9, 10, 1, 14, 15, 16, 18, 0}. (d) = {1,, 3, 4, 1, 13,..., 0}; C» ={3, 13, 17, 19}. (e) = {, 4, 6,..., 0}; C = Ø. s illustrated in the example above, the intersetion of two sets ould be empty (another reason why we need to have the empty set). The sets and are said to be disjoint if there are no elements that are in both and ; that is, if = Ø. Venn Diagrams. piture is worth a thousand words. Venn diagram provides a way to represent sets and set operations pitorially. Let be a set with universal set. The Venn diagram representing and is 6

7 Figure 1 The region inside the retangle orresponds to the elements in the universal set ; the region inside the irle are the elements of the set. Now let and be sets ontained within the same universal set. The following figures represent possible relationships between and : Figure Figure 3 Figure 4 Figure displays disjoint sets and ; the sets and in Figure 3 are not disjoint; and Figure 4 illustrates the subset relationship. Here is a subset of,. Venn diagrams provide a onvenient way to illustrate the set operations: omplement, union and intersetion. Figure 5 shows, the set of elements that are not in. Figure 5 7

8 is the set of elements that are either in or in or in both and ; is the set of elements that are in both and. These ases are shown in Figures 6 and 7. Figure 6 Figure 7 Examples.6: For eah of the following, use a opy of the Venn diagram in Figure 3 to shade the indiated region. (a) (b) () ( ) (d) ( ) Venn diagram for two sets desribes the four disjoint regions labeled R, R, R and R in Figure R 1 R 3 R R 4 Figure 8 R 1 onsists of the elements that are in but not in. Thus, R onsists of the elements that are in both and. Thus, R 3 onsists of the elements that are in but not in. Thus, R 1 =. R =. R3 =. 8

9 R 4 onsists of the elements that are not in and not in. Thus, R 4 =. Venn diagrams an also be used to illustrate the various unions, intersetions and omplements of three sets. Let, and C be sets ontained in some universal set. Venn diagram for the three sets is shown in Figure 9. C Figure 9 Notie that a Venn diagram representing three sets desribes eight distint, disjoint regions as shown in Figure 10. R R 1 R 3 R 4 R 5 R 6 R 7 R 8 C Figure 10 R 1 onsists of the elements that are in but not in and not in C. In set notation, R1 = C. R onsists of the elements that are in and but not in C. Thus, nd so on. R = C. Exerises: 1. In eah of the following, state whether or not the given olletion is a set. If it is, speify the set using the roster method. a. Distint letters in the word mathematis. b. Handsome men in Texas.. Counting numbers stritly between 10 and 0. d. Good students enrolled at Texas &M niversity. e. The odd ounting numbers. f. The positive multiples of 3. 9

10 . Write the following sets using set builder notation. a. {1, 3, 5, 7,... }. b. { 0, ± 1, ±, }. {Sunday, Monday,..., Saturday} d. {a, e, i, o, u} e. {Washington, dams, Jefferson,..., Clinton, ush} f. {101, 10, 103, 104,... } In Problems 3 and 4, let (universal set) = {0,1,,,10}, = {0,,4,6,8}, = { 1,3,5,7,10} and C = {,3,5, 6,9,10}. 3. Indiate T true or F false for eah of the following: a. b.. 3 d. { 6} e. { 1,,3,4} C f. «œ g. = {8, 0, 6,, 4} h. {1, 3, 5, 7, 9} i. {, 4, 6} 4. se the roster method to speify eah of the following sets: a. b. C. d. C e. ( ) f. 5. Let and be sets in a universal set. se opies of the Venn diagram in Figure 3 to shade eah of the following regions: a. b.. d. e. ) ( f. 6. Let, and C be sets in a universal set. se opies of the Venn diagram in Figure 9 to shade eah of the following regions: a. C b. ( C). ( ) C d. ( C) e. ( C) f. ( C) 7. Let and be sets in a universal set. What an you onlude if: a. =? b. =? 8. se opies of the Venn diagram in Figure 3 to show that: a. ( ) =. b. ( ) =. These results are known as DeMorgan s Laws after the 19 th entury mathematiian gustus DeMorgan. 10

11 1. a. { matheis,,,,,,, } b. Not a set.. {11,1,13,,19} d. Not a set. e. {1, 3,5,7, } f. { mis,,, p } g. {3, 6, 9,1, } nswers to the Exerises Chapter 1. a. S = { x x is an odd positive integer} = { x x is an odd ounting number} b. S = P{ x x is an integer}. S = { x x is a dya of the week} d. S = { x x is a vowel} e. S = { x x is a President of the nited States} f. S = { x x is a positive integer and x > 100} 3. a. T b. F. F d. T e. T f. F g. T h. T i. T 4. a. {0,1,,3,4,5,6,7,8,10} b. (,6}. {0,,4,6,8,9} d. {1, 7} e. f. 7. a. If =, then. b. If =, then. 11

SET DEFINITION 1 elements members

SET DEFINITION 1 elements members SETS SET DEFINITION 1 Unordered collection of objects, called elements or members of the set. Said to contain its elements. We write a A to denote that a is an element of the set A. The notation a A denotes