# Problem of the Week Problem A and Solution Pet Problem

## Problem

There are $$30$$ students in Ms. Chan’s class. They were discussing their pets. They were asked three questions and the results are shown in the table below:

Question Number of Students Who Answered Yes
Do you have at least one dog at home? $$16$$
Do you have at least one cat at home? $$19$$
Do you have both at least one cat and at least one dog at home? $$11$$

How many students have neither cats nor dogs at home? You may use a Venn diagram to help solve this problem.

Not printing this page? You can use the Venn diagram on our interactive worksheet.

## Solution

Since $$11$$ students have both at least one cat and at least one dog, we can put $$11$$ in the overlapping section of the Venn diagram. Now, $$16$$ students have at least one dog at home and $$11$$ have dogs and cats, so $$16-11 = 5$$ students have only dogs at home. Similarly, $$19$$ students have at least one cat at home, so $$19-11 = 8$$ students have only cats at home. The Venn diagram below summarizes this.

Now we need to add up all the pet owners.

$$5 \text{ (dogs only)} + 11 \text{ (cats and dogs)} + 8 \text{ (cats only)} = 24 \text{ pet owners}$$.

There are $$30$$ students in the class in total. Therefore the remaining $$30 - 24 = 6$$ students do not have cats nor dogs as pets.

### Teacher’s Notes

A Venn diagram is a way to visualize logical relationships among sets. Set theory is a fundamental area of mathematics. Consider two sets: $$A$$ and $$B$$. The union of $$A$$ and $$B$$, denoted by $$A \cup B$$, is the set that contains all of the elements that appear in either $$A$$ or $$B$$ (or both). The intersection of $$A$$ and $$B$$, denoted by $$A \cap B$$, is the set that contains all of the elements that appear in both $$A$$ and $$B$$. The complement of $$A$$, denoted $$\overline{A}$$, is the set that contains all the elements that do not appear in set $$A$$.

In this problem, the union is the set containing the $$24$$ students who own either a cat or a dog or both. The intersection is the set containing the $$11$$ students who own both a cat and a dog. The complement of the union is the set containing the $$6$$ students who own neither a cat nor a dog.

A Venn diagram may involve more than two sets. For example, here is a diagram of three sets, $$A$$, $$B$$, and $$C$$, that shows eight regions:

Intersection Region(s)
$$A \cap B$$ 2 and 4
$$A \cap C$$ 3 and 4
$$B \cap C$$ 4 and 6
$$A \cap \overline{B} \cap \overline{C}$$ 1
$$A \cap B \cap \overline{C}$$ 2
$$A \cap \overline{B} \cap C$$ 3
$$A \cap B \cap C$$ 4
$$\overline{A} \cap B \cap \overline{C}$$ 5
$$\overline{A} \cap B \cap C$$ 6
$$\overline{A} \cap \overline{B} \cap C$$ 7
$$\overline{A} \cap \overline{B} \cap \overline{C}$$ 8

Drawing a Venn diagram with 4 or more sets and all possible intersections gets very tricky.