2.5 Set Applications and Survey Data Analysis 81 I II III IV U Pizza 95 80 61 14 Chicken wings Figure 2.26 Solution We will use P to represent customers who purchased pizza and C to represent customers who purchased chicken wings. The example provides the following information. The number of customers surveyed is 250: = nU( ) 250. The number of customers surveyed who purchased pizza is 175: = n P( ) 175. The number of customers surveyed who purchased chicken wings is 141: = n C( ) 141. The number of customers surveyed who purchased both pizza and chicken wings is 80: n C (P ) > is 80. We illustrate this information in the Venn diagram shown in Fig. 2.26. We already know that P C> corresponds to region II. Because = n P C ( ) 80, > we write 80 in region II. Set P consists of regions I and II. We know that set P, the number of customers who purchased pizza, contains 175 customers. The total of these two regions must be 175. Region II contains 80 customers. Therefore, region I contains − = 175 80 95 customers. We write the number 95 in region I. Set C consists of regions II and III. Because = n C( ) 141, the total in these two regions must be 141. Region II contains 80, leaving − = 141 80 61 customers for region III. We write 61 in region III. The total number of customers surveyed who purchased pizza or chicken wings is determined by adding the numbers in regions I, II, and III. Therefore n C (P ) 95 < = + 80 61 236. + = The number of customers in region IV is the difference between nU( ) and n C (P ). < There are − 250 236, or 14, customers in region IV. a) The customers surveyed who did not purchase either pizza or chicken wings are those customers in the universal set who are not contained in set P or set C. The 14 customers in region IV did not purchase pizza or chicken wings. b) The 61 customers in region III are those customers surveyed who purchased chicken wings but not pizza. c) The 95 customers in region I are those customers surveyed who purchased pizza but not chicken wings. d) The customers in regions I, II, and III are those customers surveyed who purchased either pizza or chicken wings. Thus, + + = 95 80 61 236 customers surveyed purchased either pizza or chicken wings. Notice that the 80 customers in region II who purchased both pizza and chicken wings are included in those customers surveyed who purchased either pizza or chicken wings. 7 Application of Venn Diagrams Learning Catalytics Keyword: Angel-SOM-2.5 (See Preface for additional details.) Similar problems involving three sets can be solved, as illustrated in Example 2. Example 2 Software Sales Survey Software Depot surveyed 893 of its customers and obtained the following data. 545 purchased games. 497 purchased educational software. 290 purchased utility programs. 297 purchased games and educational software. 196 purchased educational software and utility programs. 205 purchased games and utility programs. 157 purchased all three types of software. Use a Venn diagram to answer the following questions. How many customers purchased a) none of these types of software? b) only games? c) at least one of these types of software? d) exactly two of these types of software? Now try Exercise 3
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