2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 75 Now try Exercise 61 Determine < > A B C ( ) Set Corresponding Regions A I, II, IV, V >B C V, VI < > A B C ( ) I, II, IV, V, VI Determine < > < A B A C ( ) ( ) Set Corresponding Regions <A B I, II, III, IV, V, VI <A C I, II, IV, V, VI, VII < > < A B A C ( ) ( ) I, II, IV, V, VI The regions that correspond to < > A B C ( ) are I, II, IV, V, and VI, and the regions that correspond to < > < A B A C ( ) ( ) are also I, II, IV, V, and VI. The results show that both statements are represented by the same regions, namely, I, II, IV, V, and VI, and therefore < > < > < A B C A B A C ( ) ( ) ( ) = for all sets A B, , and C. 7 U A B III II I IV V VI VII VIII C Figure 2.20 all sets A and B. No matter what sets we choose for A and B, this statement will be true. For example, let U A {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {3, 4, 6, 10}, = = and B {1, 2, 4, 5, 6, 8}. = ′ = ′ ′ ′ = ′ ′ = = > < < < A B A B ( ) {4, 6} {3, 4, 6, 10} {1, 2, 4, 5, 6, 8} {1, 2, 3, 5, 7, 8, 9, 10} {1, 2, 5, 7, 8, 9} {3, 7, 9, 10} {1, 2, 3, 5, 7, 8, 9, 10} {1, 2, 3, 5, 7, 8, 9, 10} We can also use Venn diagrams to prove statements involving three sets. Example 4 Equality of Sets Determine whether < > < > < A B C A B A C ( ) ( ) ( ) = for all sets, A B, , and C. Solution Because the statements include three sets, A B, , and C, three circles must be used. The Venn diagram illustrating the eight regions is shown in Fig. 2.20. First we will determine the regions that correspond to < > A B C ( ), and then we will determine the regions that correspond to < > < A B A C ( ) ( ). If both answers are the same, the statements are equal. Venn Diagrams In Example 4, we proved that < > < > < A B C A B A C ( ) ( ) ( ) = for all sets A B, , and C. Show that this statement is true for the specific sets U {1, 2, 3, 4, 5, = A B 6, 7, 8, 9, 10}, {1, 2, 3, 7}, {2, 3, 4, 5, 7, 9}, = = and C {1, 4, 7, 8, 10}. = De Morgan’s Laws for Sets In set theory, logic, and other branches of mathematics, a pair of related theorems known as De Morgan’s laws make it possible to transform statements and formulas into alternative and often more convenient forms. In set theory, De Morgan’s laws are symbolized as follows. De Morgan’s Laws for Sets 1. > < A B A B ( )′ = ′ ′ 2. < > A B A B ( )′ = ′ ′ Law 1 was verified in Example 3. We suggest that you verify law 2 at this time. The laws were expressed verbally by William of Ockham in the fourteenth century. In the nineteenth century, Augustus De Morgan expressed them mathematically. De Morgan’s laws will be discussed more thoroughly in Chapter 3, Logic. Instructor Resources for Section 2.4 in MyLab Math • Objective-Level Videos 2.4 • Interactive Concept Video: Recognizing Regions of Venn Diagrams • Animation: Venn Diagrams • PowerPoint Lecture Slides 2.4 • MyLab Exercises and Assignments 2.4
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