Freezy's Ice Cream Stand polls its customers on their favorite flavor: chocolate or vanilla? 103 customers said they liked chocolate, 98 customer said they like vanilla, while 27 customers said they liked both chocolate and vanilla. How many customers said they like only chocolate? Use a Venn diagram to help you.
Guidance
A Venn diagram is shown below.
The diagram illustrates that within some universe of data, there are two subsets,
Example A
At a school of 500 students, there are 125 students enrolled in Algebra II, 257 students who play sports and 52 students that are enrolled in Algebra II and play sports. Create a Venn diagram to illustrate this information.
Solution: First, let’s let set
There symbols that can be used to describe the number of elements in each region in the diagram as well.
Symbol  Description  Value for this Problem 


The number of elements in set 
125 

The number of elements in the intersection of sets 
52 

The number of elements in the union of sets 
330 

The number of elements in the compliment of 
375 

The number of elements in the compliment of 
170 

The number of elements in the compliment of 
448 

The number of elements in the intersection of 
73 
Example B
Create a Venn diagram to illustrate the following information regarding the subsets
Solution: Again, we will start in the middle or intersection. We must determine how many elements are in the intersection. Let’s consider that when we add the elements in
In general, for two sets,
Example C
Create a Venn diagram to represent the following information and answer the questions that follow.
In a survey of 150 high school students it was found that:
80 students have laptops
110 students have cell phones
125 students have iPods
62 students have both a laptop and a cell phone
58 students have both a laptop and iPod
98 students have both a cell phone and an iPod
50 students have all three items
a. How many students have just a cell phone?
b. How many students have none of the mentioned items?
c. How many students have an iPod and laptop but not a cellphone?
Solution: First we will use the given information to construct the Venn diagram as shown.
We can start by putting
Now that the Venn diagram is complete, we can use it to answer the questions.
a. There are 0 students that just have a cell phone.
b. There are 3 students with none of the mentioned technology.
c. There are 8 students with an iPod and laptop but no cell phone.
Intro Problem Revisit The number of customers who said they liked both chocolate and vanilla is the intersection of the two circles in the Venn diagram that represents this situation. Since a total of 103 people said they liked chocolate, we must subtract the number who like both chocolate and vanilla to find the number who like only chocolate.
Therefore, 76 of Freezy's customers said they only like chocolate ice cream.
Guided Practice
Use the Venn diagram to determine the number of elements in each set described in the problems.
1.
2.
3. \begin{align*}n(A^\prime)\end{align*}
4. \begin{align*}n(A \cap B)\end{align*}
5. \begin{align*}n(A \cup B \cup C)\end{align*}
6. \begin{align*}n(A \cap C^\prime)\end{align*}
7. \begin{align*}n(A \cap B \cap C)\end{align*}
8. \begin{align*}n(A^\prime \cap B^\prime \cap C^\prime)\end{align*}
Answers
1. \begin{align*}3 + 7 + 8 + 8 =26\end{align*}
2. \begin{align*}8 + 8 + 4 + 12 = 32\end{align*}
3. \begin{align*}8 + 4 + 12 + 6 = 30\end{align*}
4. \begin{align*}7 + 8 = 15\end{align*}
5. \begin{align*}3 + 7 + 8 + 8 + 8 + 4 + 12 = 50\end{align*}
6. \begin{align*}3 + 7 =10\end{align*}.
7. 8
8. 6
Practice
Use the information below for problems 15.
In a survey of 80 households, it was found that:
30 had at least one dog
42 had at least one cat
21 had at least one “other” pet (fish, turtle, reptile, hamster, etc.)
20 had dog(s) and cat(s)
10 had cat(s) and “other” pet(s)
8 had dog(s) and “other” pet(s)
5 had all three types of pets
 Make a Venn diagram to illustrate the results of the survey.
 How many have dog(s) and cat(s) but no “other” pet(s)?
 How many have only dog(s)?
 How many have no pets at all?
 How many “other” pet(s) owners also have dog(s) or cat(s) but not both?
Use the letters in the Venn diagram below to describe the region for each of the sets.
 \begin{align*}A \cap B\end{align*}
 \begin{align*}A\end{align*}
 \begin{align*}A \cup B\end{align*}
 \begin{align*}A \cap B^\prime\end{align*}
 \begin{align*}(A \cap B)^\prime\end{align*}
 \begin{align*}(A \cup B)^\prime\end{align*}
 \begin{align*}A^\prime\end{align*}
 \begin{align*}B^\prime \cup A\end{align*}