SOLUTION 1.3.1
The key to solving this problem is to recognize that the information refers to two categories that overlap.
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 part of set E, so we must place 3 people in the other part of set E:
We are also told that "49 believe that they have been abducted by space aliens." This means that set A must contain 49 people. Since 42 of them have already been place in one part of circle A, we must have 7 people in the other part of circle A:
Finally, we are told that 64 people were surveyed. This means that there must be a total of 64 people in this universe. So far, we have placed 52 people in three regions of the universe. Therefore, there must be 12 people in the region that is outside of the two circles:
Now that we have organized the given information so that there is one number in each of the four regions of the Venn diagram, we can can use the diagram to answer the questions.
1. How many believe neither of these things?
If a person believes neither of these things, then the person isn't in set E and isn't in set A. The diagram shows us that 12 people satisfy this description.
2. How many believe Elvis is still alive but don't believe that they have been abducted by space aliens?
A person who fits this description is simultaneously inside of circle E yet outside of circle A. The diagram shows us that there are 3 of these people.