This problem involves using the principle of inclusion-exclusion to find out how many people own only a bird, only a cat, or neither.

Given Information:

• Total number of surveyed people: $132$
• People who own a bird: $66$
• People who own a cat: $85$
• People who own both a bird and a cat: $47$

Questions to Answer:

1. How many people own only a bird?
2. How many people own only a cat?
3. How many people own either a bird or a cat (or both)?
4. How many people do not own either a bird or a cat?

Solutions:

1. Number of people who own only a bird:

   To find the number of people who own only a bird, subtract the number of people who own both a bird and a cat from the number of people who own a bird: $\text{People who own only a bird} = 66 - 47 = 19$

2. Number of people who own only a cat:

   Similarly, subtract the number of people who own both a bird and a cat from the number of people who own a cat: $\text{People who own only a cat} = 85 - 47 = 38$

3. Number of people who own either a bird or a cat (or both):

   Use the principle of inclusion-exclusion to find the number of people who own either a bird or a cat (or both): $\text{People who own either or both} = 66 + 85 - 47 = 104$

4. Number of people who do not own either a bird or a cat:

   Subtract the number of people who own either a bird or a cat (or both) from the total number of surveyed people: $\text{People who do not own either} = 132 - 104 = 28$

Summary:

• People who own only a bird: 19
• People who own only a cat: 38
• People who own either a bird or a cat (or both): 104
• People who do not own either: 28

Would you like further clarification on any part of this solution or have any more questions?

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Relative Questions:

1. How many people own exactly one pet (either a bird or a cat, but not both)?
2. What percentage of the surveyed people own both a bird and a cat?
3. If 10 more people owned both a bird and a cat, how would the calculations change?
4. How many more people would need to own only a cat to have an equal number of cat-only and bird-only owners?
5. How would the results change if the total number of surveyed people increased to 200?

Tip:

When solving problems involving sets and overlapping categories, the principle of inclusion-exclusion is a powerful tool for accurately counting elements in various categories.