Venn diagram showing sets P and Q with elements 2, 8, 3, 12, 15, 11, 7, 10, and 16, alongside questions on set operations and matching diagrams to set notation.

Problem Analysis:

The task involves working with sets using a Venn diagram and understanding set operations such as union, intersection, complement, and their combinations. Let's break down each part of the problem step by step.

#### Given Venn Diagram:
- Set \( P \): Elements = {1, 2, 8, 10, 12}
- Set \( Q \): Elements = {3, 7, 11, 15, 16}
- Intersection \( P \cap Q \): Elements = {3, 12}

#### Tasks:
1. Identify elements in \( P \cup Q \)
2. Identify elements in \( P \cap Q \)
3. Identify elements in \( P' \) (complement of \( P \))
4. Identify elements in \( (P \cap Q)' \) (complement of the intersection of \( P \) and \( Q \))

#### Matching Venn Diagrams with Set Notation:
We need to match the given Venn diagrams with the correct set notation:
- \( A' \cap B \)
- \( (A \cap B)' \)
- \( A \cap B \)
- \( A \cup B \)

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Solution:

#### Task (i): Identify elements in \( P \cup Q \)
The union of two sets \( P \cup Q \) includes all elements that are in either \( P \), \( Q \), or both.

From the Venn diagram:
- Elements in \( P \): {1, 2, 8, 10, 12}
- Elements in \( Q \): {3, 7, 11, 15, 16}
- Elements in \( P \cap Q \): {3, 12} (already counted in both \( P \) and \( Q \))

Combining all unique elements:
\[ P \cup Q = \{1, 2, 3, 7, 8, 10, 11, 12, 15, 16\} \]

Answer for (i): \( \{1, 2, 3, 7, 8, 10, 11, 12, 15, 16\} \)

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#### Task (ii): Identify elements in \( P \cap Q \)
The intersection of two sets \( P \cap Q \) includes only the elements that are common to both \( P \) and \( Q \).

From the Venn diagram:
- Elements in \( P \cap Q \): {3, 12}

Answer for (ii): \( \{3, 12\} \)

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#### Task (iii): Identify elements in \( P' \)
The complement of a set \( P' \) includes all elements in the universal set that are not in \( P \).

Assuming the universal set \( U \) contains all elements in the Venn diagram:
\[ U = \{1, 2, 3, 7, 8, 10, 11, 12, 15, 16\} \]

Elements in \( P \): {1, 2, 8, 10, 12}

Complement \( P' \):
\[ P' = U - P = \{3, 7, 11, 15, 16\} \]

Answer for (iii): \( \{3, 7, 11, 15, 16\} \)

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#### Task (iv): Identify elements in \( (P \cap Q)' \)
The complement of the intersection \( (P \cap Q)' \) includes all elements in the universal set that are not in \( P \cap Q \).

From the Venn diagram:
- Elements in \( P \cap Q \): {3, 12}
- Universal set \( U \): {1, 2, 3, 7, 8, 10, 11, 12, 15, 16}

Complement \( (P \cap Q)' \):
\[ (P \cap Q)' = U - (P \cap Q) = \{1, 2, 7, 8, 10, 11, 15, 16\} \]

Answer for (iv): \( \{1, 2, 7, 8, 10, 11, 15, 16\} \)

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Matching Venn Diagrams with Set Notation:

1. \( A' \cap B \):
- This represents the elements that are in \( B \) but not in \( A \).
- Matched Venn diagram: The region outside \( A \) but inside \( B \).

2. \( (A \cap B)' \):
- This represents the elements that are not in both \( A \) and \( B \).
- Matched Venn diagram: The entire universal set minus the intersection of \( A \) and \( B \).

3. \( A \cap B \):
- This represents the elements that are in both \( A \) and \( B \).
- Matched Venn diagram: The overlapping region of \( A \) and \( B \).

4. \( A \cup B \):
- This represents the elements that are in either \( A \), \( B \), or both.
- Matched Venn diagram: The entire region covered by \( A \) and \( B \).

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Final Answers:

1. \( P \cup Q = \{1, 2, 3, 7, 8, 10, 11, 12, 15, 16\} \)
2. \( P \cap Q = \{3, 12\} \)
3. \( P' = \{3, 7, 11, 15, 16\} \)
4. \( (P \cap Q)' = \{1, 2, 7, 8, 10, 11, 15, 16\} \)

\[
\boxed{\{1, 2, 3, 7, 8, 10, 11, 12, 15, 16\}, \{3, 12\}, \{3, 7, 11, 15, 16\}, \{1, 2, 7, 8, 10, 11, 15, 16\}}
\]