Question

7 of the people on Hansen's street have a lizard and 9 have a cat. 6 people have both a lizard and a cat. How many people have a lizard or a cat or both?

Hint: Copy and complete the Venn diagram below to help you solve the problem.

​
I have a lizard
I have a cat

people

Answer 1

To solve this problem, we can use the principle of inclusion-exclusion. We have the following information:

- Let \( A \) be the set of people who have a lizard.
- Let \( B \) be the set of people who have a cat.

From the problem:

- \( |A| = 7 \) (people who have a lizard)
- \( |B| = 9 \) (people who have a cat)
- \( |A \cap B| = 6 \) (people who have both a lizard and a cat)

We want to find the total number of people who have either a lizard or a cat or both, which is represented by \( |A \cup B| \).

Using the inclusion-exclusion principle:

\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

Substituting the values we have:

\[
|A \cup B| = 7 + 9 - 6
\]

Calculating this gives:

\[
|A \cup B| = 16 - 6 = 10
\]

So, the total number of people who have either a lizard or a cat or both is **10**.

### Venn Diagram Breakdown

Here’s a breakdown you can visualize in a Venn diagram context:

- People with only a lizard (not a cat): \( |A| - |A \cap B| = 7 - 6 = 1 \)
- People with only a cat (not a lizard): \( |B| - |A \cap B| = 9 - 6 = 3 \)
- People with both a lizard and a cat: \( |A \cap B| = 6 \)

In the Venn diagram:

- In the "I have a lizard" circle, you have **1** (only lizard) + **6** (both lizard and cat) = 7.
- In the "I have a cat" circle, you have **3** (only cat) + **6** (both lizard and cat) = 9.

Thus, adding these distinct groups gives us the total of **10** individuals who have either a lizard, a cat, or both.