A and B are two sets containing repectively `m_(1)andm_(2)` elements.
If `xlen(AuuB)ley`, find x and y.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that:

\[ x \leq n(A \cup B) \leq y \]

where \( n(A) = m_1 \) and \( n(B) = m_2 \).

### Step 1: Understanding the Union of Sets
The number of elements in the union of two sets \( A \) and \( B \) can be expressed using the formula:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

This means that the number of elements in the union is equal to the sum of the number of elements in both sets minus the number of elements that are in both sets (the intersection).

### Step 2: Finding the Maximum Value of \( n(A \cup B) \)
To find the maximum value of \( n(A \cup B) \), we need to minimize \( n(A \cap B) \). The minimum value of \( n(A \cap B) \) is \( 0 \) (when the sets do not overlap). Thus, we have:

\[ n(A \cup B) \leq n(A) + n(B) = m_1 + m_2 \]

So, in this case, \( y = m_1 + m_2 \).

### Step 3: Finding the Minimum Value of \( n(A \cup B) \)
To find the minimum value of \( n(A \cup B) \), we need to maximize \( n(A \cap B) \). The maximum value of \( n(A \cap B) \) is limited by the size of the smaller set. Therefore, we have two cases based on the sizes of the sets:

1. **Case 1**: If \( m_1 \geq m_2 \), then the maximum intersection is \( m_2 \):
\[ n(A \cup B) \geq n(A) + n(B) - n(A \cap B) = m_1 + m_2 - m_2 = m_1 \]
Thus, in this case, \( x = m_1 \).

2. **Case 2**: If \( m_2 > m_1 \), then the maximum intersection is \( m_1 \):
\[ n(A \cup B) \geq n(A) + n(B) - n(A \cap B) = m_1 + m_2 - m_1 = m_2 \]
Thus, in this case, \( x = m_2 \).

### Step 4: Finalizing the Values of \( x \) and \( y \)
Now we can summarize the results:

- If \( m_1 \geq m_2 \):
\[ m_1 \leq n(A \cup B) \leq m_1 + m_2 \]
Therefore, \( x = m_1 \) and \( y = m_1 + m_2 \).

- If \( m_2 > m_1 \):
\[ m_2 \leq n(A \cup B) \leq m_1 + m_2 \]
Therefore, \( x = m_2 \) and \( y = m_1 + m_2 \).

### Conclusion
In conclusion, the values of \( x \) and \( y \) are:
- \( x = \min(m_1, m_2) \)
- \( y = m_1 + m_2 \)