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I see the image contains several mathematical problems related to set theory, limits, and measure theory. Here's a brief overview of the questions visible:

1. **Question 3:**  
   Given \( A_1 = \{(x,y) \mid 0 < x < 2, 1 < y < 2\} \) and \( A_2 = \{(x,y) \mid 1 < x < 3, 1 < y < 3\} \), determine \( A_1 \cup A_2 \) and \( A_1 \cap A_2 \).

2. **Question 4:**  
   Given \( \Omega = \{(x,y) \mid x^2 + y^2 \leq 2\} \) and \( A = \{(x,y) \mid y \leq x\} \), determine \( A^c \) (the complement of A).

3. **Question 5:**  
   Given \( \Omega = \{1, 2, 3, 4\} \) and a collection \( A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{2,4\}, \Omega \} \), investigate if \( A \) forms a **σ-field** or not.

4. **Question 6:**  
   With \( \Omega = \{1, 2, 3, 4\} \) and \( A = \{ \emptyset, \{1,2\}, \{3,4\}, \Omega \} \), investigate whether \( A \) forms a **σ-field** or not.

5. **Question 7:**  
   If \( \Omega = \{1, 2, 3, 4\} \) and \( C = \{\{1\}, \{2\}\} \), determine \( \sigma(C) \) (the σ-field generated by C).

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Would you like detailed solutions to any specific questions? 

Here are some follow-up questions for expanding your understanding:

1. What are the conditions for a collection of sets to form a **σ-field**?
2. How do you compute the union and intersection of regions defined by inequalities?
3. What is the complement of a set in terms of geometric regions?
4. How do you generate the **σ-field** from a collection of sets?
5. What are the concepts of **lim inf** and **lim sup** in the context of sequences of sets?

**Tip:** When determining if a collection is a **σ-field**, check if it satisfies closure under complementation and countable unions.

This image contains several mathematical problems related to set theory, limits, and measure theory. Problems involve operations on sets, determining sigma-fields, and calculating limit superior (lim sup) and limit inferior (lim inf) of sequences of sets.

Explore set theory, sigma-fields, and limits of sets with these advanced mathematical problems. This set of problems includes operations on sets, determining sigma-fields, and calculating lim sup and lim inf of sequences of sets. Suitable for university students in advanced mathematics courses.

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