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2 THE NATURE OF SETS Copyright © Cengage Learning. All rights reserved. 2.1 Sets, Subsets, and Venn Diagrams Copyright © Cengage Learning. All rights reserved. Denoting Sets 3 Denoting Sets “Set: a collection of objects.” What is a collection? “Collection: an accumulation.” What is an accumulation? “Accumulation: a collection, a pile, or a heap.” We see that the word collection gives us a circular definition. What is a pile? “Pile: a heap.” What is a heap? “Heap: a pile.” 4 Denoting Sets Sets are usually specified in one of two ways. The first is by description, and the other is by the roster method. In the description method, we specify the set by describing it in such a way that we know exactly which elements belong to it. An example is the set of 50 states in the United States of America. We say that this set is well defined. 5 Denoting Sets The distinctive property that determines the inclusion or exclusion of a particular element is called the defining property of the set. In the roster method, the set is defined by listing the members. The objects in a set are called members or elements of the set and are said to belong to or be contained in the set. For example, instead of defining a set as the set of all students in this class who received a C or better on the first examination, we might simply define the set by listing its members: {Howie, Mary, Larry}. 6 Denoting Sets Sets are usually denoted by capital letters, and the notation used for sets is braces. Thus, the expression A = {4, 5, 6} means that A is the name for the set whose members are the numbers 4, 5, and 6. Sometimes we use braces with a defining property, as in the following examples: {states in the United States of America} {all students in this class who received an A on the first test} 7 Denoting Sets If S is a set, we write a S if a is a member of the set S, and we write b S if b is not a member of the set S. Thus, “a ” means that the variable a is an integer, and the statement “b , b 0” means that the variable b is a nonzero integer. 8 Example 1 – Set member notation Let C = cities in California, a = city of Anaheim, and b = city of Berlin. Use set membership notation to describe relations among a, b, and C. Solution: a C; b C 9 Denoting Sets A common use of set terminology is to refer to certain sets of numbers. Some examples of set are listed below. {1, 2, 3, 4, . . .} Set of natural, or counting, numbers {0, 1, 2, 3, 4, . . .} Set of whole numbers {. . . , –2, –1, 0, 1, 2, . . .} Set of integers Set of rational numbers 10 Denoting Sets A new notation called set-builder notation was invented to allow us to combine both the roster and the description methods. Consider: 11 Denoting Sets We now use set-builder notation for the set of rational numbers: { | a is an integer and b is a nonzero integer} Read this as: “The set of all b is a nonzero integer.” such that a is an integer and 12 Example 2 – Writing sets by roster Specify the given sets by roster. If the set is not well defined, say so. a. {counting numbers between 10 and 20} b. {x | x is an integer between –20 and 20} Solution: a. {11, 12, 13, 14, 15, 16, 17, 18, 19} Notice that between does not include the first and last numbers. 13 Example 2 – Solution cont’d b. {–19, –18, –17, ...,17, 18, 19} Notice that ellipses (three dots) are used to denote some missing numbers. When using ellipses, you must be careful to list enough elements so that someone looking at the set can see the intended pattern. 14 Example 3 – Writing sets by description Specify the given sets by description. a. {1, 2, 3, 4, 5, ...} b. {0, 1, 2, 3, 4, 5, ...} c. {x | x } d. {12, 14, 16, ..., 98} e. {4, 44, 444, 4444, . . .} f. {m, a, t, h, e, i, c, s} 15 Example 3 – Solution Answers may vary. a. Counting (or natural) numbers b. Whole numbers c. We would read this as “The set of all x such that x belongs to the set of integers.” More simply, the answer is integers. d. {Even numbers between 10 and 100} e. {Counting numbers whose digits consist of fours only} f. {Distinct letters in the word mathematics} 16 Equal and Equivalent Sets 17 Equal and Equivalent Sets We say that two sets A and B are equal, written A = B, if the sets contain exactly the same elements. Thus, if E = {2, 4, 6, 8, . . .}, then {x | x is an even counting number} = {x | x E} The order in which you represent elements in a set has no effect on set membership. Thus, {1, 2, 3} = {3, 1, 2} = {2, 1, 3} = . . . 18 Equal and Equivalent Sets Also, if an element appears in a set more than once, it is not generally listed more than a single time. For example, {1, 2, 3, 3} = {1, 2, 3} Another possible relationship between sets is that of equivalence. Two sets A and B are equivalent, written A B, if they have the same number of elements. Equivalent sets do not need to be equal sets, but equal sets are always equivalent. 19 Example 4 – Equal and equivalent sets Which of the following sets are equivalent? Are any equal? Solution: All of the given sets are equivalent. Notice that no two of them are equal, but they all share the property of “fourness.” 20 Equal and Equivalent Sets The number of elements in a set is often called its cardinality. The cardinality of the sets in Example 4 is 4; that is, the common property of the sets is the cardinal number of the set. The cardinality of a set S is denoted by |S|. Equivalent sets with four elements each have in common the property of “fourness,” and thus we would say that their cardinality is 4. 21 Example 5 – Finding cardinality Find the cardinality of each of the following sets. a. b. S = { } c. T = {states of the United States} 22 Example 5 – Solution a. The cardinality of R is 4, so we write |R| = 4. b. The cardinality of S (the empty set) is 0, so we write |S| = 0. c. The cardinality of T is 50, or |T| = 50. 23 Universal and Empty Sets 24 Universal and Empty Sets We now consider two important sets in set theory. The first is the set that contains every element under consideration, and the second is the set that contains no elements. A universal set, denoted by U, contains all the elements and the empty set contains no elements, and thus has cardinality 0. The empty set is denoted by { } or Do not confuse the notations . , 0, and { }. 25 Universal and Empty Sets The symbol denotes a set with no elements; the symbol 0 denotes a number; and the symbol { } is a set with one element (namely, the set containing ) For example, if U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, then all sets we would be considering would have elements only among the elements of U. No set could contain the number 10, since 10 is not in that agreed-upon universe. 26 Universal and Empty Sets A universal set must be specified or implied, and it must remain fixed for that problem. However, when a new problem is begun, a new universal set can be specified. The following are examples of descriptions of the empty set: {living saber-toothed tigers} {counting numbers less than 1} 27 Venn Diagrams 28 Venn Diagrams A set is a collection, and a useful way to depict a set is to draw a circle or an oval as a representation for the set. The elements are depicted inside the circle, and objects not in the set are shown outside the circle. The universal set contains all the elements under consideration in a given discussion and is depicted as a rectangle. This representation of a set is called a Venn diagram, after John Venn (1834–1923). 29 Example 6 – Venn diagram for a given set Let the universal set be all of the cards in a deck of cards. Draw a Venn diagram for the set of hearts. Solution: It is customary to represent the universal set as a rectangle (labeled U) and the set of hearts (labeled H) as a circle, as shown in Figure 2.1. Venn diagram for a deck of cards Figure 2.1 30 Example 6 – Solution cont’d Note that In the Venn diagram, the sets involved are too large to list all of the elements individually in either H or U, but we can say that the two of hearts (labeled ) is a member of H, whereas the two of diamonds (labeled ) is not a member of H. We write H, whereas H 31 Venn Diagrams The set of elements that are not in H is referred to as the complement of H, and this is written using an overbar. In Example 6, H = {spades, diamonds, clubs}. A Venn diagram for complement is shown in Figure 2.2. Venn diagrams for a set and for its complement Figure 2.2 32 Venn Diagrams Notice that any set S divides the universe into two regions as shown in Figure 2.3. General representation of a set S Figure 2.3 33 Venn Diagrams Notice that the cardinality of the deck of cards in Example 6 is 52, and the cardinality of H is 13. Since the deck of cards is the universal set for Example 6, we can symbolize the cardinality of these sets as follows: |H| = 13 and |U| = 52 Note that H |H|. In words, a set is not the same as its cardinality. 34 Subsets and Proper Subsets 35 Subsets and Proper Subsets Most applications will involve more than one set, so we begin by considering the relationships between two sets A and B. The various possible relationships are shown in Figure 2.4. Relationships between two sets A and B Figure 2.4 36 Subsets and Proper Subsets We say that A is a subset of B, which in set theory is written A B, if every element of A is also an element of B (see Figure 2.4a). Similarly, B A if every element of B is also an element of A (Figure 2.4b). Figure 2.4c shows two equal sets. Finally, A and B are disjoint if they have no elements in common (Figure 2.4d). 37 Example 8 – Subsets of a given set Find all possible subsets of C = {5, 7}. Solution: {5}, {7} are obvious subsets. {5, 7} is also a subset, since both 5 and 7 are elements of C. { } is also a subset of C. It is a subset because all of its elements belong to C. 38 Example 8 – Solution cont’d Stated a different way, if it were not a subset of C, we would have to be able to find an element of { } that is not in C. Since we cannot find such an element, we say that the empty set is a subset of C (and, in fact, the empty set is a subset of every set). 39 Subsets and Proper Subsets The subsets of C can be classified into two categories: proper and improper. Since every set is a subset of itself, we immediately know one subset for any given set: the set itself. A proper subset is a subset that is not equal to the original set; that is, A is a proper subset of a subset B, written A B, if A is a subset of B and A ≠ B. An improper subset of a set A is the set A. We see there are three proper subsets of C = {5,7} : and {7}. There is one improper subset of C: {5, 7}. , {5} 40 Example 9 – Classifying proper and improper subsets Find the proper and improper subsets of A = {2, 4, 6, 8}. What is the cardinality of A? Solution: The cardinality of A is 4 (because there are 4 elements in A). There is one improper subset: {2, 4, 6, 8}. The proper subsets are as follows: { }, {2}, {4}, {6}, {8}, {2, 4}, {2, 6}, {2, 8}, {4, 6}, {4, 8}, {6, 8}, {2, 4, 6}, {2, 4, 8}, {2, 6, 8}, {4, 6, 8} 41 Subsets and Proper Subsets 42 Subsets and Proper Subsets Sometimes we are given two sets X and Y, and we know nothing about the way they are related. In this situation, we draw a general figure, such as the one shown in Figure 2.5. General Venn diagram for two sets Figure 2.5 43 Example 10 – Regions in a Venn diagram Name the regions in Figure 2.6 described by each of the following. a. A b. C c. A d. B e. A B f. A and C are disjoint General Venn diagram for three sets Figure 2.6 44 Example 10 – Solution a. A is regions I, IV, V, and VII. b. C is regions III, IV, VI, and VII. c. A is regions II, III, VI, and VIII. d. B is regions I, III, IV, and VIII. e. A B means that regions I and IV are empty. f. A and C are disjoint means that regions IV and VII are empty. 45