Learning Objectives
- Applications (surveys) of Venn diagrams with 2 sets
- Applications (surveys) of Venn diagrams with 3 sets
Applications of Venn Diagrams with Two Sets
EXAMPLE
For example, suppose you gave a survey and found that 15 people liked only dogs, 10 people liked only cats, and 40 people liked both dogs and cats. The Venn diagram would consist of two circles, one labeled dogs and one labeled cats.
Notice that if you look at the set of those who like dogs, you will see there are 15 who like dogs only, and 40 who like both. The total number of people who like dogs is the sum of 15 and 40, which is 55.
Likewise, if you look at the set of those who like cats, you will see there are 10 who like cats only, and 40 who like both. The total number of people who like cats is the sum of 40 and 10, which is 50.
Exercises
A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:
- Tea only
- Coffee only
- Both coffee and tea
Suppose 20 report tea only, 80 report coffee only, 40 report both. How many people drink tea in the morning? How many people drink neither tea or coffee?
Try It
Example
Suppose you asked 100 people if they like Star Wars or Star Trek.
- 32 like both
- 40 liked just Star Wars
- 20 liked just Star Trek
- 8 liked neither
We can fill in a Venn diagram with this information. First, start with the general format for two sets:
Then, fill in the number in the intersection.
Fill in the “just Star Wars”, “just Star Trek”, and neither areas.
Example
Another, slightly harder example. Suppose you conducted a survey and asked if people liked Popeye’s chicken sandwich or Wendy’s chicken sandwich. You found:
12 liked both
30 liked Popeye’s
14 liked Wendy’s
Start with your blank Venn Diagram.
Next, like before, fill in the intersection.
The wording was a little different in this example. Note, we know the TOTAL number of people who liked Popeye’s. We will need to subtract off the overlap in order to find out how many people liked just Popeye’s. So, 30-12=18. Put this in your diagram.
Do the same thing with Wendy’s, subtract off the intersection to find out how many people just like Wendy’s.
Finally, add all of the numbers in the circles and overlap. This is how many people answered that they liked either Popeye’s or Wendy’s or both. Subtract this number from the total surveyed. We have 50-32=18. Put this in the diagram outside of the circles.
Example
Suppose you surveyed a group of 160 people who ordered pizza from Pizza Hut and asked them what they ordered.
150 ordered pizza
80 ordered bread sticks
90 ordered soda
50 ordered pizza and bread sticks (no soda)
70 ordered pizza and soda (no bread sticks)
0 ordered bread sticks and soda (no pizza)
20 ordered all three
Start in the middle and work your way out. So start with 20 who ordered all three.
Next, add in where two sets overlap.
Next, you want to find out how many ordered JUST pizza, JUST soda, and JUST bread sticks. We know that 150 ordered pizza, so take 150 and subtract off the numbers already in the pizza circle (50, 20, and 70). You should get 10. Do this with soda and bread sticks as well, and add your results to the Venn diagram.
Finally, you want to figure out if there is anyone in the universal set (the 160 asked) who did not order pizza, bread sticks, or soda. You want to add all the numbers in the circles and subtract this from 160. 160- (10+50+20+70+10+0+0)= 160-160=0
We do not have any members who need to be accounted for outside of the circles.
This is the end of the section. Close this tab and proceed to the corresponding assignment.
