For the universal set, \( U=\{5,6,7,8,9\} \), complete the parts below. Write your answers in roster form or as \( \varnothing \). (a) Suppose \( A=\{5,7,8,9\} \). Then what is \( A^{\prime} \) ? \[ A^{\prime}=\square \] (b) Suppose we know that \( B^{\prime}=\{6,8\} \). Then what would \( B \) have to be? \[ B=\square \]

Let's solve each part step by step. ### Part (a) **Given:** - Universal set: \( U = \{5, 6, 7, 8, 9\} \) - Set \( A = \{5, 7, 8, 9\} \) **To find:** - The complement of \( A \), denoted \( A^{\prime} \), which includes all elements in \( U \) that are not in \( A \). **Solution:** \[ A^{\prime} = U \setminus A = \{5, 6, 7, 8, 9\} \setminus \{5, 7, 8, 9\} = \{6\} \] **Answer:** \[ A^{\prime} = \{6\} \] ### Part (b) **Given:** - Universal set: \( U = \{5, 6, 7, 8, 9\} \) - The complement of set \( B \), denoted \( B^{\prime} = \{6, 8\} \) **To find:** - The set \( B \), which includes all elements in \( U \) that are not in \( B^{\prime} \). **Solution:** \[ B = U \setminus B^{\prime} = \{5, 6, 7, 8, 9\} \setminus \{6, 8\} = \{5, 7, 9\} \] **Answer:** \[ B = \{5, 7, 9\} \]