72 CHAPTER 2 Sets Example 1 illustrates the general procedure. U A B I II III IV V VI 1 8 VII VIII C 12 10 2 4 5 9 6 7 3 11 Figure 2.17 GENERAL PROCEDURE FOR CONSTRUCTING VENN DIAGRAMS WITH THREE SETS, A,B, AND C 1. Determine the elements to be placed in region V by determining the elements that are common to all three sets, > > A B C. 2. Determine the elements to be placed in region II. Determine the elements in >A B. The elements in this set belong in regions II and V. Place the elements in the set >A B that are not listed in region V in region II. The elements in regions IV and VI are determined in a similar manner. 3. Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are determined in a similar manner. 4. Determine the elements to be placed in region VIII by determining the elements in the universal set that are not in regions I through VII. PROCEDURE U A B I II III IV VI V VIII VII A ù B ù C Region V C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C U A B I II III IV VI V VIII VII C A ù B Regions II, V A ù C Regions IV, V B ù C Regions V, VI A ø B Regions I, II, III, IV, V, VI A ø C Regions I, II, IV, V, VI, VII (A ø B ø C)' Region VIII B ø C Regions II, III, IV, V, VI, VII A ø B ø C Regions I, II, III, IV, V, VI, VII Figure 2.16 Example 1 Constructing a Venn Diagram for Three Sets Construct a Venn diagram illustrating the following sets. U A B C {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} {5, 9, 10, 12} {1, 2, 4, 5, 8, 9, 10} {l, 3, 5, 8, 9, 11} = = = = Solution First determine the intersection of all three sets. Because the elements 5 and 9 are in all three sets, > > A B C {5, 9}. = The elements 5 and 9 are placed in region V in Fig. 2.17. Next complete region II by determining the intersection of sets A and B. >A B {5, 9, 10} = >A B consists of regions II and V. The elements 5 and 9 have already been placed in region V, so 10 must be placed in region II.
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