2.3 Venn Diagrams, Set Operations, and Data Representation 65 The Number of Elements in <A B For any finite sets A and B, = + − nAB nAnBnAB ( ) ( ) ( ) ( ) < > Example 9 How Many Visitors Speak Spanish or French? The data obtained from of a survey of visitors at the Grand Canyon showed that 25 speak Spanish, 14 speak French, and 4 speak both Spanish and French. How many speak Spanish or French? Solution If we let set A be the set of visitors who speak Spanish and let set B be the set of visitors who speak French, then we need to determine n A B ( ). < We can use the above formula to determine n A B ( ). < nAB nAnBnAB n A B ( ) ( ) ( ) ( ) ( ) 25 14 4 35 < > < = + − = + − = Thus, 35 of the visitors surveyed speak either Spanish or French. 7 Now try Exercise 89 Example 10 The Number of Elements in Set A The data obtained from a survey of customers at a Shake Shack restaurant showed that 28 purchased either a hamburger or french fries, 20 purchased french fries, and 17 purchased both a hamburger and french fries. How many customers purchased only a hamburger? Solution If we let set A be the set of customers who purchased a hamburger and set B be the set of customers who purchased french fries, we need to determine n A( ). We are given the number of customers who purchased either a hamburger or french fries, which is n A B ( ). < We are also given the number of customers who purchased french fries, n B( ), and the number of customers who purchased both a hamburger and french fries, n A B ( ). > We can use the formula = + − nAB nAnBnAB ( ) ( ) ( ) ( ) < > to solve for n A( ). nAB nAnBnAB n A n A n A n A ( ) ( ) ( ) ( ) 28 ( ) 20 17 28 ( ) 3 28 3 ( ) 3 3 25 ( ) < > = + − = + − = + − = + − = Thus, the number of customers who purchased only a hamburger is 25. 7 Now try Exercise 91 Two other set operations are the difference of two sets and the Cartesian product. We will first discuss the difference of two sets. Difference of Two Sets Definition: Difference of Two Sets The difference of two sets A and B, symbolized − A B, is the set of elements that belong to set A but not to set B. m The Grand Canyon Joshua Hartmann/Shutterstock
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