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.

### Union and Intersection of Sets

A Venn diagram is shown below.

The diagram illustrates that within some universe of data, there are two subsets,

Let's solve the following problems.

- 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.

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 |

- Create a Venn diagram to illustrate the following information regarding the subsets
M andN in universeU :

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,

- 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

- How many students have just a cell phone?
- How many students have none of the mentioned items?
- How many students have an iPod and laptop but not a cellphone?

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.

- There are 0 students that just have a cell phone.
- There are 3 students with none of the mentioned technology.
- There are 8 students with an iPod and laptop but no cell phone.

### Examples

#### Example 1

Earlier, you were asked to find the number of customers that said they only like chocolate.

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.

**Use the Venn diagram below to determine the number of elements in each set described in the following examples.**

#### Example 2

#### Example 3

\begin{align*}n(C)\end{align*}

\begin{align*}8 + 8 + 4 + 12 = 32\end{align*}

#### Example 4

\begin{align*}n(A^\prime)\end{align*}

\begin{align*}8 + 4 + 12 + 6 = 30\end{align*}

#### Example 5

\begin{align*}n(A \cap B)\end{align*}

\begin{align*}7 + 8 = 15\end{align*}

#### Example 6

\begin{align*}n(A \cup B \cup C)\end{align*}

\begin{align*}3 + 7 + 8 + 8 + 8 + 4 + 12 = 50\end{align*}

#### Example 7

\begin{align*}n(A \cap C^\prime)\end{align*}

\begin{align*}3 + 7 =10\end{align*}

#### Example 8

\begin{align*}n(A \cap B \cap C)\end{align*}

8

#### Example 9

\begin{align*}n(A^\prime \cap B^\prime \cap C^\prime)\end{align*}

6

### Review

Use the information below for problems 1-5.

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*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 12.12.