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The image contains 14 questions on set theory, combinatorics, and probability. Here are the solutions with explanations for each question:

### Solutions:

1. **Q1. Write the roster form of \( H = \{ x | x \in \mathbb{N} \text{ and } 5 < x^2 < 50 \} \).**
   - Find integers \( x \) such that \( 5 < x^2 < 50 \):
     - Possible values of \( x \): \( x = 3, 4, 5, -3, -4, -5 \)
     - So, \( H = \{3, 4, 5, -3, -4, -5\} \).

2. **Q2. Give an example of two sets whose intersection is an empty set.**
   - Example: \( A = \{1, 2\} \) and \( B = \{3, 4\} \).
   - \( A \cap B = \emptyset \).

3. **Q3. How many subsets a set has containing 5 elements.**
   - A set with \( n \) elements has \( 2^n \) subsets.
   - If \( n = 5 \): total subsets \( = 2^5 = 32 \).

4. **Q4. Write the set \( \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \} \) in set builder form.**
   - Answer: \( S = \left\{ \frac{1}{n} \;|\; n \in \mathbb{N} \right\} \).

5. **Q5. Define power set.**
   - The power set of a set \( A \) is the set of all subsets of \( A \), including \( A \) itself and the empty set.

6. **Q6. Write all the subsets of the set \( \{1, 2\} \).**
   - Subsets of \( \{1, 2\} \): \( \{\}, \{1\}, \{2\}, \{1, 2\} \).

7. **Q7. Find sets \( A, B, C \) such that \( A \cap B, B \cap C, A \cap C \) are non-empty sets, and \( A \cap B \cap C = \emptyset \).**
   - Example: \( A = \{1, 2\} \), \( B = \{2, 3\} \), \( C = \{1, 3\} \).

8. **Q8. Two finite sets having \( m \) and \( k \) elements. If the total number of subsets of the first set is 56 more than the total number of subsets of the second set, find \( m \) and \( k \).**
   - \( 2^m = 2^k + 56 \).
   - Solve for integers \( m \) and \( k \).

9. **Q9. For all sets \( A, B, C \), is \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)? Justify your answer.**
   - Yes, by distributive laws.

10. **Q10. If \( U = \{1, 2, 3, \dots, 10\} \), \( A = \{x \;|\; x \text{ is prime}\} \), \( B = \{x \;|\; x \text{ is even integer}\} \), find \( A \cup B^c \).**

11. **Q11. \( A \) and \( B \) are two sets such that \( n(A - B) = 14 + x \), \( n(B - A) = 3x \), and \( n(A \cap B) = x \).**


The image contains a series of 14 questions related to set theory, combinatorics, and probability, covering topics such as set notation, subsets, power sets, set operations, and basic probability.

Explore a series of 14 questions on set theory, combinatorics, and probability, including concepts like subsets, power sets, and set operations. Suitable for high school students in Grades 11-12, this page provides detailed solutions and explanations for each question.

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