If B subseteq A, A cap B^prime=1,4,5 and A cup B=1,2,3,4,5,6 , then find set B .

## Step 1: Understand the Given Sets<br />### We are given that \( B \subseteq A \), \( A \cap B^{\prime} = \{1,4,5\} \), and \( A \cup B = \{1,2,3,4,5,6\} \). Here, \( B^{\prime} \) represents the complement of \( B \) with respect to \( A \).<br />## Step 2: Determine Set \( A \)<br />### Since \( A \cap B^{\prime} = \{1,4,5\} \), these elements are in \( A \) but not in \( B \). Therefore, \( A \) must include these elements. Additionally, since \( B \subseteq A \), all elements of \( B \) are also in \( A \).<br />## Step 3: Use Union Information<br />### The union \( A \cup B = \{1,2,3,4,5,6\} \) implies that all elements of both sets \( A \) and \( B \) are included in this set. Thus, \( A \) must contain all elements from \( A \cap B^{\prime} \) and any additional elements needed to satisfy the union.<br />## Step 4: Find Set \( B \)<br />### Since \( B \subseteq A \) and \( A \cap B^{\prime} = \{1,4,5\} \), the elements \( 1, 4, \) and \( 5 \) cannot be in \( B \). The remaining elements \( 2, 3, \) and \( 6 \) must be in \( B \) to satisfy \( A \cup B = \{1,2,3,4,5,6\} \).