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  3. TRAVELERS SURVEYED: 250 In a survey of 250 European travelers, 93 have traveled to Africa, 155 have traveled to Asia, and of th

TRAVELERS SURVEYED: 250 In a survey of 250 European travelers, 93 have traveled to Africa, 155 have traveled to Asia, and of th

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问题 TRAVELERS SURVEYED: 250

In a survey of 250 European travelers, 93 have traveled to Africa, 155 have traveled to Asia, and of these two groups, 70 have traveled to both continents, as illustrated in the Venn diagram above.
(a)How many of the travelers surveyed have traveled to Africa but not to Asia?
(b)How many of the travelers surveyed have traveled to at least one of the two continents of Africa and Asia?
(c)How many of the travelers surveyed have traveled neither to Africa nor to Asia?
选项
答案23 178 72
解析 In the Venn diagram, the rectangular region represents the set of all travelers surveyed; the two circular regions represent the two sets of travelers to Africa and Asia, respectively; and the shaded region represents the subset of those who have traveled to both continents.
(a)The set described here is represented by the part of the left circle that is not shaded. This description suggests that the answer can be found by taking the shaded part away from the first circle—in effect, subtracting the 70 from the 93, to get 23 travelers who have traveled to Africa but not to Asia.
(b)The set described here is represented by that part of the rectangle that is in at least one of the two circles. This description suggests adding the two numbers 93 and 155. But the 70 travelers who have traveled to both continents would be counted twice in the sum 93 + 155. To correct the double counting, subtract 70 from the sum so that these 70 travelers are counted only once:
93 + 155-70=178
(c)The set described here is represented by the part of the rectangle that is not in either circle. Let N be the number of these travelers. Note that the entire rectangular region has two main nonoverlapping parts: the part outside the circles and the part inside the circles. The first part represents N travelers and the second part represents 93 + 155 - 70 = 178 travelers(from question(b)). Therefore,
250 = N+178
and solving for N yields
N= 250 -178 = 72
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