(i) If A and B sets have 2 and 5 elements respectively, then find the minimum and maximum number of elements in `A cup B`
(ii) If A and B have 4 and 6 elements respectively, then find the minimum and maximum number of elements in `A cap B`.

Answer

To solve the given question, we will break it down into two parts as specified.

### Part (i): Finding Minimum and Maximum of \( A \cup B \)

1. **Identify the number of elements in sets A and B:**
- Let \( |A| = 2 \) (number of elements in set A)
- Let \( |B| = 5 \) (number of elements in set B)

2. **Use the formula for the union of two sets:**
\[ |A \cup B| = |A| + |B| - |A \cap B| \]

3. **Determine the maximum value of \( |A \cap B| \):**
- The maximum number of common elements (intersection) cannot exceed the number of elements in the smaller set, which is set A.
- Thus, \( |A \cap B|_{\text{max}} = 2 \).

4. **Calculate the minimum number of elements in \( A \cup B \):**
\[ |A \cup B|_{\text{min}} = |A| + |B| - |A \cap B|_{\text{max}} = 2 + 5 - 2 = 5 \]

5. **Determine the minimum value of \( |A \cap B| \):**
- The minimum number of common elements can be zero (if there are no common elements).
- Thus, \( |A \cap B|_{\text{min}} = 0 \).

6. **Calculate the maximum number of elements in \( A \cup B \):**
\[ |A \cup B|_{\text{max}} = |A| + |B| - |A \cap B|_{\text{min}} = 2 + 5 - 0 = 7 \]

**Final Results for Part (i):**
- Minimum number of elements in \( A \cup B = 5 \)
- Maximum number of elements in \( A \cup B = 7 \)

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### Part (ii): Finding Minimum and Maximum of \( A \cap B \)

1. **Identify the number of elements in sets A and B:**
- Let \( |A| = 4 \) (number of elements in set A)
- Let \( |B| = 6 \) (number of elements in set B)

2. **Use the formula for the intersection of two sets:**
\[ |A \cap B| = |A| + |B| - |A \cup B| \]

3. **Determine the maximum value of \( |A \cup B| \):**
- The maximum number of elements in the union occurs when all elements are distinct.
- Thus, \( |A \cup B|_{\text{max}} = |A| + |B| = 4 + 6 = 10 \).

4. **Calculate the minimum number of elements in \( A \cap B \):**
\[ |A \cap B|_{\text{min}} = |A| + |B| - |A \cup B|_{\text{max}} = 4 + 6 - 10 = 0 \]

5. **Determine the minimum value of \( |A \cup B| \):**
- The minimum number of elements in the union occurs when there is maximum overlap.
- If all elements of A are also in B, then \( |A \cup B|_{\text{min}} = 6 \) (the number of elements in B).

6. **Calculate the maximum number of elements in \( A \cap B \):**
\[ |A \cap B|_{\text{max}} = |A| + |B| - |A \cup B|_{\text{min}} = 4 + 6 - 6 = 4 \]

**Final Results for Part (ii):**
- Minimum number of elements in \( A \cap B = 0 \)
- Maximum number of elements in \( A \cap B = 4 \)

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