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 $n(A \cap B) = x$.