Venn Diagram and Excellent Party Reputation

PART 1 MODULE 3 VENN DIAGRAMS AND SURVEY PROBLEMS EXAMPLE 1.3.1 A survey of 64 informed voters revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things 1. How many believe neither of these things? 2. How many believe Elvis is still alive but don't believe that they have been abducted by space aliens? SOLUTION TO EXAMPLE 1.3.1 When we first read the data in this example, it may seem as if the numbers contradict one another. For instance, we were told that 64 people were surveyed, yet there are 45 who believe that Elvis is alive and 49 who believe that they've been kidnapped by space aliens. Obviously, 45 + 49 is much greater than 64, so it appears that the number of people who responded to the survey is greater than the number of people who were surveyed. This apparent contradiction is resolved, however, when we take into account the fact that there are some people who fall into both categories ("42 believe both of those things"). A Venn diagram is useful in organizing the information in this type of problem. Since the data refers to two categories, we will use a two-circle diagram. Let U be the set of people who were surveyed. Let E be the set of people who believe that Elvis is still alive. Let A be the set of people who believe that they have been abducted by space aliens. Then we have the following Venn diagram showing the relationship between sets E, A and U:

We are told that there are 42 people who "believe both of these things." This means that in the region of the diagram where set E intersects set A, we have 42 people:

We are also told that "45 believe that Elvis is still alive." This means that set E must contain a total of 45 people. We have already placed 42 people in one region of set E, so we must place 3 people in the other region of set E:

We are also told that "49 believe that they have been abducted by space aliens." This means that set