82 CHAPTER 2 Sets U Games 102 I 200 48 IV II III Educational Utility VI VII 46 39 140 157 161 VIII V programs Figure 2.27 Timely Tip When constructing a Venn diagram, remember to subtract the number in region V from the respective values in determining the numbers to be placed in regions II, IV, and VI. Solution Begin by constructing a Venn diagram with three overlapping circles. One circle represents games, another educational software, and the third utility programs, See Fig. 2.27. Label the eight regions. Whenever possible, work from the center of the diagram outward. First fill in region V. Since 157 customers purchased all three types of software, we place 157 in region V. Next determine the number to be placed in region II. Regions II and V together represent the customers who purchased both games and educational software. Since 297 customers purchased both of these types of software, the sum of the numbers in these regions must be 297. Since 157 have already been placed in region V, − = 297 157 140 must be placed in region II. Now we determine the number to be placed in region IV. Since 205 customers purchased both games and utility programs, the sum of the numbers in regions IV and V must be 205. Since 157 customers have already been placed in region V, − = 205 157 48 must be placed in region IV. Now determine the number to be placed in region VI. A total of 196 customers purchased educational software and utility programs. The numbers in regions V and VI must total 196. Since 157 customers have already been placed in region V, the number to be placed in region VI is − = 196 157 39. Now that we have determined the numbers for regions V, II, IV, and VI, we can determine the numbers to be placed in regions I, III, and VII. We are given that 545 customers purchased games. The sum of the numbers in regions I, II, IV, and V must be 545. To determine the number to be placed in region I, subtract the amounts in regions II, IV, and V from 545. There must be − − − = 545 140 48 157 200 in region I. Determine the numbers to be placed in regions III and VII in a similar manner. Region III 497 140 157 39 161 Region VII 290 48 157 39 46 = − − − = = − − − = Now that we have determined the numbers in regions I through VII, we can determine the number to be placed in region VIII. Adding the numbers in regions I through VII yields a sum of 791. The difference between the total number of customers surveyed, 893, and the sum of the numbers in regions I through VII must be placed in region VIII. = − = Region VIII 893 791 102 Now that we have completed the Venn diagram, we can answer the questions. a) One hundred two customers did not purchase any of these types of software. These customers are indicated in region VIII. b) Region I represents those customers who purchased only games. Thus, 200 customers purchased only games. c) The words at least one mean “one or more.” All those customers in regions I through VII purchased at least one of the types of software. The sum of the numbers in regions I through VII is 791, so 791 customers purchased at least one of the types of software. d) The customers in regions II, IV, and VI purchased exactly two of the types of software. Summing the numbers in these regions + + 140 48 39 we determine that 227 customers purchased exactly two of these types of software. Notice that we did not include the customers in region V. Those customers purchased all three types of software. 7 The procedure to work problems like those given in Example 2 is generally the same. Start by completing region V. Next complete regions II, IV, and VI. Then complete regions I, III, and VII. Finally, complete region VIII. When you are constructing Venn diagrams, be sure to check your work carefully. Now try Exercise 5
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