EXAMPLE 1.3.3

A survey of faculty and graduate students at the University of Florida's film

school revealed the following information:

51 admire Moe

49 admire Larry

60 admire Curly

34 admire Moe and Larry

32 admire Larry and Curly

36 admire Moe and Curly

24 admire all three of the Stooges

1 admires none of the Three Stooges

a) How many people were surveyed?

b) How many admire Curly, but not Larry nor Moe?

c) How many admire Larry or Curly?

d) How many admire exactly one of the Stooges?

e) How many admire exactly two of the Stooges?

We will organize the information in the following Venn diagram, where "M," "L," and "C" represent the sets of those who admire Moe, Larry and Curly, respectively:

"24 admire all three of the Stooges:"

"1 admires none of the Three Stooges:"

"36 admire Moe and Curly:"

"32 admire Larry and Curly:

"34 admire Moe and Larry:"

"60 admire Curly:"

"51 admire Moe"

"49 admire Larry"

Now that we have one number in each of the diagram's eight regions, we use the numbers to answer the given questions.

a) How many people were surveyed?

We add all eight numbers.

5 + 10 + 7 + 12 + 24 + 8 + 16 + 1 = 83

b) How many admire Curly, but not Larry nor Moe?

These are the ones who are simultaneously inside of circle C yet outside of the other two circles. The diagram shows that the answer is "16."

c) How many admire Larry or Curly?

Unless we specify otherwise, we use the word "or" in the inclusive sense, so that this means "admire Larry, or admire Curly, or admire both." Those who satisfy this compound condition are underlined in the diagram below.

10 + 7 + 24 + 8 + 12 + 16 = 77

d) How many admire exactly one of the Stooges?

There are three possibilities: admires Moe but not Curly and not Larry, admires Larry but not Curly and not More, or admires Curly but not Moe and not Larry. Those who satisfy this compound condition are underlined in the diagram below.

5 + 7 + 16 = 28

e) How many admire exactly two of the Stooges?

Again, there are three possiblities: admires Moe and Larry but not Curly, admires Moe and Curly but not Larry, or admires Larry and Curly but not Moe. Those who satisfy this compound conditions are in the regions underlined below:

12 + 10 + 8 = 30