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### 1. Write down all possible subsets of the set {1,2}.
The set {1,2} has 4 possible subsets, which are:
- ∅ (empty set)
- {1}
- {2}
- {1,2}

### 3. Write the number of proper subsets of a set having six elements.
The total number of subsets of a set with \( n \) elements is \( 2^n \). For \( n = 6 \), the total number of subsets is:
\[
2^6 = 64
\]
The number of *proper* subsets is always 1 less than the total number of subsets (since the set itself is not included in proper subsets), so the number of proper subsets is:
\[
64 - 1 = 63
\]

### 10. Find the domain and range of the function \( f(x) = \frac{1}{x-1} \).
- **Domain**: The function \( f(x) = \frac{1}{x-1} \) is undefined when \( x = 1 \) because division by zero is not allowed. Hence, the domain is all real numbers except \( x = 1 \), which is:
  \[
  \text{Domain} = (-\infty, 1) \cup (1, \infty)
  \]
- **Range**: The function can take any real value except 0, because the function approaches zero but never actually reaches it. Therefore, the range is:
  \[
  \text{Range} = (-\infty, 0) \cup (0, \infty)
  \]

### 16. A and B are two sets such that \( n(A) = 3 \) and \( n(B) = 6 \). Find:
1. **Minimum value of \( n(A \cup B) \)**: The minimum value occurs when all elements of A are also in B. In this case, \( n(A \cup B) = n(B) = 6 \).
2. **Maximum value of \( n(A \cap B) \)**: The maximum value occurs when A and B have no common elements. In this case, \( n(A \cup B) = n(A) + n(B) = 3 + 6 = 9 \).

### 18. If \( B \subseteq A \), show that \( (C - B) \subseteq (C - A) \).
- Since \( B \subseteq A \), every element of \( B \) is also an element of \( A \). Therefore, any element that is in \( C - B \) (i.e., in \( C \) but not in \( B \)) will also be in \( C - A \) because \( A \) contains \( B \). Thus, \( C - B \subseteq C - A \).

Would you like details on any specific problem or more clarification?

Here are 5 related questions:
1. How many subsets would a set with 5 elements have?
2. What is the difference between proper subsets and improper subsets?
3. Can you find the range of the function \( f(x) = \frac{1}{x^2 - 4} \)?
4. What are the subsets of the set {a, b, {c}}?
5. What is the minimum value of \( n(A \cup B) \) if \( n(A) = 4 \) and \( n(B) = 5 \)?

**Tip**: Always remember that a set with \( n \) elements has \( 2^n \) total subsets, including both proper and improper ones!

Explore solutions for Class 11 Mathematics questions, covering topics like subsets of sets, proper subsets calculations, and finding the domain and range of functions. Learn key set theory concepts and solve problems related to unions and intersections. Suitable for Grade 11 students.

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