Revision quiz focusing on set operations and Venn diagrams.

Problem Analysis and Solution

The task involves solving set theory problems, including listing the members of sets, finding intersections, and unions. Let's solve each part step by step.

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#### Part 1: Sets A and B

##### Step 1: Define Set A
Set \( A \) is defined as the first 4 square numbers.
- The first 4 square numbers are: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \).
- Therefore, \( A = \{1, 4, 9, 16\} \).

##### Step 2: Define Set B
Set \( B \) is defined as the composite numbers less than 12.
- Composite numbers are numbers greater than 1 that are not prime (i.e., they have divisors other than 1 and themselves).
- The composite numbers less than 12 are: \( 4, 6, 8, 9, 10 \).
- Therefore, \( B = \{4, 6, 8, 9, 10\} \).

##### Step 3: Find \( A \cup B \) (Union of A and B)
The union of two sets includes all elements that are in either set.
- \( A = \{1, 4, 9, 16\} \)
- \( B = \{4, 6, 8, 9, 10\} \)
- Combining all unique elements: \( A \cup B = \{1, 4, 6, 8, 9, 10, 16\} \).

##### Step 4: Find \( A \cap B \) (Intersection of A and B)
The intersection of two sets includes only the elements that are common to both sets.
- \( A = \{1, 4, 9, 16\} \)
- \( B = \{4, 6, 8, 9, 10\} \)
- The common elements are: \( 4 \) and \( 9 \).
- Therefore, \( A \cap B = \{4, 9\} \).

##### Final Answers for Part 1
\[
\begin{aligned}
A &= \{1, 4, 9, 16\} \\
B &= \{4, 6, 8, 9, 10\} \\
A \cup B &= \{1, 4, 6, 8, 9, 10, 16\} \\
A \cap B &= \{4, 9\}
\end{aligned}
\]

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#### Part 2: Sets P and Q

##### Step 1: Define Set P
Set \( P \) is given as: \( P = \{a, e, i, o, u\} \).

##### Step 2: Define Set Q
Set \( Q \) is given as: \( Q = \{a, b, c, d, e\} \).

##### Step 3: Find \( P \cup Q \) (Union of P and Q)
The union of two sets includes all elements that are in either set.
- \( P = \{a, e, i, o, u\} \)
- \( Q = \{a, b, c, d, e\} \)
- Combining all unique elements: \( P \cup Q = \{a, b, c, d, e, i, o, u\} \).

##### Step 4: Find \( P \cap Q \) (Intersection of P and Q)
The intersection of two sets includes only the elements that are common to both sets.
- \( P = \{a, e, i, o, u\} \)
- \( Q = \{a, b, c, d, e\} \)
- The common elements are: \( a \) and \( e \).
- Therefore, \( P \cap Q = \{a, e\} \).

##### Final Answers for Part 2
\[
\begin{aligned}
P \cup Q &= \{a, b, c, d, e, i, o, u\} \\
P \cap Q &= \{a, e\}
\end{aligned}
\]

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#### Part 3: Venn Diagram Analysis

The Venn diagram shows two sets \( X \) and \( Y \) with the following regions:
- Region 3: Elements only in \( X \)
- Region 4: Elements only in \( Y \)
- Region 2: Elements in both \( X \) and \( Y \)
- Region 5: Elements only in \( X \)
- Region 6: Elements only in \( Y \)
- Region 7: Elements only in \( X \)
- Region 8: Elements only in \( Y \)

##### Step 1: Identify Elements in Each Region
From the diagram:
- Region 3: Elements only in \( X \)
- Region 4: Elements only in \( Y \)
- Region 2: Elements in both \( X \) and \( Y \)
- Region 5: Elements only in \( X \)
- Region 6: Elements only in \( Y \)
- Region 7: Elements only in \( X \)
- Region 8: Elements only in \( Y \)

##### Step 2: Find \( A \cup B \) (Union of X and Y)
The union of \( X \) and \( Y \) includes all elements in either set.
- Elements in \( X \): Regions 3, 5, 2, 7
- Elements in \( Y \): Regions 4, 6, 2, 8
- Combining all unique elements: \( A \cup B = \{3, 4, 5, 6, 7, 8, 2\} \).

##### Step 3: Find \( A \cap B \) (Intersection of X and Y)
The intersection of \( X \) and \( Y \) includes only the elements that are in both sets.
- Elements in both \( X \) and \( Y \): Region 2.
- Therefore, \( A \cap B = \{2\} \).

##### Final Answers for Part 3
\[
\begin{aligned}
A \cup B &= \{3, 4, 5, 6, 7, 8, 2\} \\
A \cap B &= \{2\}
\end{aligned}
\]

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Final Answer Summary

\[
\boxed{
\begin{aligned}
1. & \quad A = \{1, 4, 9, 16\}, \quad B = \{4, 6, 8, 9, 10\}, \quad A \cup B = \{1, 4, 6, 8, 9, 10, 16\}, \quad A \cap B = \{4, 9\} \\
2. & \quad P \cup Q = \{a, b, c, d, e, i, o, u\}, \quad P \cap Q = \{a, e\} \\
3. & \quad A \cup B = \{3, 4, 5, 6, 7, 8, 2\}, \quad A \cap B = \{2\}
\end{aligned}
}
\]