Question

• 664 people were surveyed in all
• 388 of the people own a cat
• 331 of the people own a bird
• 210 of the people own both a cat and a bird
• 103 of the people own both a cat and a fish
• 119 of the people own both a bird and a fish
• 67 of the people own all three types of pets (cat, bird, fish)
• 43 of the people dont own any of these three types of pets (cat, bird, fish)

construct a venn diagram illustrating these results. then answer the question.
image: venn diagram with three circles (own a cat, own a bird, own a fish) and partial question: \how many of t but dont own a 143 people\

Answer:

To fill the Venn diagram, we use the principle of inclusion - exclusion for three - set (cat, bird, fish) problems. Let \(C\) represent the set of people who own a cat, \(B\) represent the set of people who own a bird, and \(F\) represent the set of people who own a fish.

Step 1: Find the number of people who own only cat and bird (not fish)

We know that the number of people who own both a cat and a bird is \(n(C\cap B)=210\), and the number of people who own all three pets is \(n(C\cap B\cap F) = 67\).
The number of people who own only a cat and a bird (but not a fish) is given by:
\(n(C\cap B)-n(C\cap B\cap F)=210 - 67=143\)

Step 2: Find the number of people who own only cat and fish (not bird)

We know that the number of people who own both a cat and a fish is \(n(C\cap F)=103\), and the number of people who own all three pets is \(n(C\cap B\cap F)=67\).
The number of people who own only a cat and a fish (but not a bird) is given by:
\(n(C\cap F)-n(C\cap B\cap F)=103 - 67 = 36\)

Step 3: Find the number of people who own only bird and fish (not cat)

We know that the number of people who own both a bird and a fish is \(n(B\cap F)=119\), and the number of people who own all three pets is \(n(C\cap B\cap F) = 67\).
The number of people who own only a bird and a fish (but not a cat) is given by:
\(n(B\cap F)-n(C\cap B\cap F)=119 - 67=52\)

Step 4: Find the number of people who own only a cat

We know that the number of people who own a cat is \(n(C) = 388\). The number of people who own a cat and at least one other pet (bird or fish or both) is \(n((C\cap B)\cup(C\cap F))=n(C\cap B)+n(C\cap F)-n(C\cap B\cap F)\)
\(n((C\cap B)\cup(C\cap F))=210 + 103-67=246\)
The number of people who own only a cat is \(n(C)-n((C\cap B)\cup(C\cap F))=388 - 246 = 142\)

Step 5: Find the number of people who own only a bird

We know that the number of people who own a bird is \(n(B)=331\). The number of people who own a bird and at least one other pet (cat or fish or both) is \(n((B\cap C)\cup(B\cap F))=n(B\cap C)+n(B\cap F)-n(C\cap B\cap F)\)
\(n((B\cap C)\cup(B\cap F))=210+119 - 67 = 262\)
The number of people who own only a bird is \(n(B)-n((B\cap C)\cup(B\cap F))=331-262 = 69\)

Step 6: Find the number of people who own only a fish

First, we find the number of people who own at least one of the three pets. The total number of people surveyed is \(N = 664\), and the number of people who own none of the three pets is \(43\). So the number of people who own at least one of the three pets is \(664 - 43=621\)

The number of people who own at least one of the three pets can also be calculated using the formula for three - set union:
\(n(C\cup B\cup F)=n(C)+n(B)+n(F)-n(C\cap B)-n(C\cap F)-n(B\cap F)+n(C\cap B\cap F)\)
We know \(n(C) = 388\), \(n(B)=331\), \(n(C\cap B)=210\), \(n(C\cap F)=103\), \(n(B\cap F)=119\), \(n(C\cap B\cap F)=67\) and \(n(C\cup B\cup F) = 621\)

We can solve for \(n(F)\):
\(621=388 + 331+n(F)-210-103 - 119+67\)
\(621=388+331-210 - 103-119 + 67+n(F)\)
\(388+331=719\), \(210 + 103+119=432\), \(719-432 + 67=354\)
\(621=354+n(F)\)
\(n(F)=621 - 354=267\)

The number of people who own a fish and at least one other pet (cat or bird or both) is \(n((F\cap C)\cup(F\cap B))=n(F\cap C)+n(F\cap B)-n(C\cap B\cap F)=103 + 119-67 = 155\)
The number of people who own only a fish is \(n(F)-n((F\cap C)\cup(F\cap B))=267-155 = 112\)

Step 7: Number of people who own none of the three pets

We are given that the number of people who don't own any of these three types of pets is \(43\)

Now, we can fill the Venn diagram:

• Only cat: \(142\)
• Only bird: \(69\)
• Only fish: \(112\)
• Cat and bird only: \(143\)
• Cat and fish only: \(36\)
• Bird and fish only: \(52\)
• All three (cat, bird, fish): \(67\)
• None of the three: \(43\)

If we assume the question (since part of the question is cut off, but from the given "143 people" and our calculation of cat and bird only) is about the number of people who own a cat and a bird but not a fish, the answer is \(143\)

If we were to fill all the boxes:

• The box in the "Own a cat" circle only: \(142\)
• The box in the intersection of "Own a cat" and "Own a bird" only (not including "Own a fish"): \(143\)
• The box in the "Own a bird" circle only: \(69\)
• The box in the intersection of all three circles: \(67\)
• The box in the intersection of "Own a cat" and "Own a fish" only (not including "Own a bird"): \(36\)
• The box in the intersection of "Own a bird" and "Own a fish" only (not including "Own a cat"): \(52\)
• The box in the "Own a fish" circle only: \(112\)
• The box outside all three circles (don't own any of the three): \(43\)