74 CHAPTER 2 Sets U A B I II III IV Figure 2.19 Example 3 Equality of Sets Determine whether > < A B A B ( )′ = ′ ′ for all sets A and B. Solution Draw a Venn diagram with two sets A and B, as in Fig. 2.19. Label the regions as indicated. Determine >A B ( )′ Set Corresponding Regions A I, II B II, III >A B II >A B ( )′ I, III, IV Determine < A B ′ ′ Set Corresponding Regions A′ III, IV B′ I, IV < A B ′ ′ I, III, IV Both statements are represented by the same regions, I, III, IV, of the Venn diagram. Thus, > < A B A B ( )′ = ′ ′ for all sets A and B. 7 Now try Exercise 51 Learning Catalytics Keyword: Angel-SOM-2.4 (See Preface for additional details.) Verification of Equality of Sets In this chapter, for clarity we may refer to operations on sets, such as <A B′ or > > A B C, as statements involving sets or simply as statements. Now we discuss how to determine if two statements involving sets are equal. Consider the question: Is < > A B A B ′ = ′ for all sets A and B? For the specific sets U A {1, 2, 3, 4, 5}, {1, 3}, = = and B {2, 4, 5}, = is < > A B A B? ′ = ′ To answer the question, we do the following. Determine : A B ′ Determine A B "′ < A B A B {2, 4, 5} {2, 4, 5} {2, 4, 5} ′ = = ′ = > A B A B {2, 4, 5} {2, 4, 5} {2, 4, 5} ′ = = ′ = For these sets, < > A B A B, ′ = ′ because both set statements are equal to {2, 4, 5}. At this point you may believe that < > A B A B ′ = ′ for all sets A and B. If we select the sets U A {1, 2, 3, 4, 5}, {1, 3, 5}, = = and B {2, 3}, = we see that < A B {2, 3, 4} ′ = and > A B {2}. ′ = For this case, < > A B A B. ′ ≠ ′ Thus, we have proved that < > A B A B ′ ≠ ′ for all sets A and B by using a counterexample. A counterexample, as explained in Chapter 1, is an example that shows a statement is not true. In Chapter 1, we explained that proofs involve the use of deductive reasoning. Recall that deductive reasoning begins with a general statement and works to a specific conclusion. To verify, or determine whether set statements are equal for any two sets selected, we use deductive reasoning with Venn diagrams. Venn diagrams are used because they can illustrate general cases. To determine if statements that contain sets, such as >A B ( )′ and < A B, ′ ′ are equal for all sets A and B, we use the regions of Venn diagrams. If both statements represent the same regions of the Venn diagram, then the statements are equal for all sets A and B. We demonstrate this process in Example 3. In Example 3, when we proved that > < A B A B ( ) , ′ = ′ ′ we started with two general sets and worked to the specific conclusion that both statements represented the same regions of the Venn diagram. We showed that > < A B A B ( )′ = ′ ′ for
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