Question

own a dog but not a bird
own a bird but not a dog
own a dog or a bird (or both)
number of people
143
29
172
construct a venn diagram illustrating these results. then answer the questions.
how many people own a dog?
how many people own both a dog and a bird?

Explanation:

Part 1: How many people own a dog?

Step1: Recall the formula for set union

The number of people who own a dog or a bird (or both) is given by \( n(D \cup B) = n(D) + n(B) - n(D \cap B) \), but we can also find \( n(D) \) as the number of people who own a dog but not a bird plus the number of people who own both a dog and a bird. We know \( n(D \text{ but not } B) = 143 \), and we will find \( n(D \cap B) \) in the next part. But first, let's use the formula for \( n(D \cup B) \). We know \( n(D \cup B) = 172 \), \( n(B \text{ but not } D) = 29 \). Let \( x = n(D \cap B) \). Then \( n(D) = 143 + x \), \( n(B) = 29 + x \). And \( n(D \cup B) = n(D) + n(B) - n(D \cap B) = (143 + x) + (29 + x) - x = 172 + x \). But we know \( n(D \cup B) = 172 \), so \( 172 + x = 172 \)? Wait, no, that's not right. Wait, actually, the number of people who own a dog is the number who own a dog but not a bird plus the number who own both. Let's find the number who own both first.

Step2: Find the number of people who own both (for the second part, but we can use it here)

Wait, let's first solve the second part: How many people own both a dog and a bird?

We know that \( n(D \cup B) = n(D \text{ only}) + n(B \text{ only}) + n(D \cap B) \). So \( 172 = 143 + 29 + n(D \cap B) \). Then \( 172 = 172 + n(D \cap B) \), so \( n(D \cap B) = 172 - 143 - 29 = 0 \)? Wait, that can't be. Wait, no, the table says: Own a dog but not a bird: 143, Own a bird but not a dog: 29, Own a dog or a bird (or both): 172. So using the formula \( n(D \cup B) = n(D \text{ only}) + n(B \text{ only}) + n(D \cap B) \), so \( 172 = 143 + 29 + n(D \cap B) \). Then \( 143 + 29 = 172 \), so \( 172 = 172 + n(D \cap B) \), so \( n(D \cap B) = 0 \). Wait, that means no one owns both? Then the number of people who own a dog is \( n(D \text{ only}) + n(D \cap B) = 143 + 0 = 143 \)? But that seems odd. Wait, maybe I misread the table. Let's check again:

The table has:

• Own a dog but not a bird: 143

• Own a bird but not a dog: 29

• Own a dog or a bird (or both): 172

So by the principle of inclusion - exclusion, \( n(D \cup B) = n(D) + n(B) - n(D \cap B) \), and also \( n(D) = n(D \text{ only}) + n(D \cap B) \), \( n(B) = n(B \text{ only}) + n(D \cap B) \). So substituting, \( n(D \cup B) = (n(D \text{ only}) + n(D \cap B)) + (n(B \text{ only}) + n(D \cap B)) - n(D \cap B) = n(D \text{ only}) + n(B \text{ only}) + n(D \cap B) \). So \( 172 = 143 + 29 + n(D \cap B) \), so \( n(D \cap B) = 172 - 143 - 29 = 0 \). So the number of people who own a dog is \( n(D \text{ only}) + n(D \cap B) = 143 + 0 = 143 \).

Wait, but that seems strange. Alternatively, maybe the table is:

• Own a dog but not a bird: 143

• Own a bird but not a dog: 29

• Own a dog or a bird (or both): 172

So the number of people who own a dog is the number who own a dog but not a bird plus the number who own both. Since \( n(D \cup B) = 172 \), and \( n(B \text{ only}) = 29 \), then the number of people who own a dog (including those who own both) is \( n(D \cup B) - n(B \text{ only}) = 172 - 29 = 143 \). Oh, right! Because \( n(D \cup B) = n(D) + n(B \text{ only}) \), since \( n(D) \) includes those who own both, and \( n(B \text{ only}) \) is those who own only birds. So \( n(D) = n(D \cup B) - n(B \text{ only}) = 172 - 29 = 143 \). Wait, that's the same as the number who own a dog but not a bird. That means no one owns both a dog and a bird. So the number of people who own a dog is 143.

Part 2: How many people own both a dog and a bird?

Step1: Use the formula for the union of two sets

We know that \( n(D \cup B) = n(D \text{ only}) + n(B \text{ only}) + n(D \cap B) \). We are given \( n(D \cup B) = 172 \), \( n(D \text{ only}) = 143 \), and \( n(B \text{ only}) = 29 \).

Step2: Substitute the values into the formula

Substituting, we get \( 172 = 143 + 29 + n(D \cap B) \).

Step3: Solve for \( n(D \cap B) \)

First, calculate \( 143 + 29 = 172 \). Then \( 172 = 172 + n(D \cap B) \), so subtracting 172 from both sides, we get \( n(D \cap B) = 172 - 172 = 0 \).

Answer:

s:

• How many people own a dog? \(\boldsymbol{143}\)

• How many people own both a dog and a bird? \(\boldsymbol{0}\)