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This problem involves a Venn diagram with four sets: fish owners (A), bird owners (B), cat owners (C), and dog owners (D). We need to determine how many pet owners have **none of these pets** (fish, birds, cats, or dogs). Let's break down the provided information:

- Total pet owners surveyed = 132
- 45 own fish
- 53 own birds
- 50 own cats
- 64 own dogs
- 11 own only fish
- 16 own only birds
- 10 own fish and birds
- 21 own fish and cats
- 26 own birds and dogs
- 29 own cats and dogs
- 3 own fish, birds, and cats
- 1 owns fish, birds, dogs, and no cats
- 9 own fish, cats, dogs, and no birds
- 10 own birds, cats, dogs, and no fish
- 11 own all four pets (fish, birds, cats, and dogs)

To find the number of people who own **none of these pets**, we will apply the principle of inclusion-exclusion:

\[
N(\text{none}) = N(\text{total}) - N(\text{fish, bird, cat, or dog})
\]

We need to calculate the number of people who own at least one of the four types of pets. Using the inclusion-exclusion principle for 4 sets:

\[
N(A \cup B \cup C \cup D) = N(A) + N(B) + N(C) + N(D) - N(A \cap B) - N(A \cap C) - N(A \cap D) - N(B \cap C) - N(B \cap D) - N(C \cap D) + N(A \cap B \cap C) + N(A \cap B \cap D) + N(A \cap C \cap D) + N(B \cap C \cap D) - N(A \cap B \cap C \cap D)
\]

Where \( A, B, C, D \) represent fish, bird, cat, and dog ownership respectively. After calculating this, we can subtract the result from 132 to find the number of people with no pets.

Let me calculate this for you.It seems that the number calculated using inclusion-exclusion is incorrect, leading to a negative result. This suggests either a mistake in the interpretation of the data or an inconsistency in the problem. Let me recheck the details and go through the process more carefully to correct this.

I'll simplify the steps and approach it again systematically.It seems that the negative result persists, indicating an inconsistency in the data provided in the problem. The negative number suggests that the sum of pet owners with overlapping categories exceeds the total number of pet owners surveyed (132), which shouldn't happen.

Would you like to review the data for possible errors, or should I explain an alternative approach to dealing with such problems in case of inconsistencies?

Let me know if you'd like more details or help with something specific. Here are five relative questions to explore:

1. How can inclusion-exclusion principles be applied to larger datasets?
2. What might cause inconsistent data in survey results like these?
3. How do you account for overcounting in problems with overlapping sets?
4. Can you solve similar problems using probability?
5. How can Venn diagrams help visualize set relationships?

**Tip:** When using inclusion-exclusion, always ensure the data provided for intersections does not exceed the totals for individual sets.

A survey of 132 pet owners yielded the following information: 45 pet owners own fish; 53 own a bird; 50 own a cat; 64 own a dog; 2 own all four; 11 own only fish; 16 own only a bird; 10 own fish and a bird; 21 own fish and a cat; 26 own a bird and a dog; 29 own a cat and a dog; 3 own fish, a bird, a cat, and no dog; 1 owns fish, a bird, a dog, and no cat; 9 own fish, a cat, a dog, and no bird; and 10 own a bird, a cat, a dog, and no fish. How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs? (They own other types of pets).

Learn how to solve a complex pet ownership survey problem using the inclusion-exclusion principle. This problem involves calculating the number of people who have no pets based on overlapping sets. Suitable for students in Grades 11-12.

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