For the universal set, $U=\{3,4,5,6,7,8,9\}$, complete the parts below. Write your answers in roster form or as $\varnothing$. (a) Suppose we know that $A^{\prime}=\{3,5,7\}$. Then what would $A$ have to be? \[ A=\square \] (b) Suppose $B=\{3,5,7,8,9\}$. Then what is $B^{\prime}$ ? \[ B^{\prime}=\square \]

Step 1 ：The universal set U is given as \(U = \{3, 4, 5, 6, 7, 8, 9\}\).

Step 2 ：For part (a), we are given that the complement of set A, denoted as \(A'\), is \(A' = \{3, 5, 7\}\). The complement of a set is the set of all elements in the universal set that are not in the given set. Therefore, to find set A, we need to find all elements in the universal set U that are not in \(A'\).

Step 3 ：By comparing the elements of the universal set U with the elements of \(A'\), we find that the elements 4, 6, 8, and 9 are not in \(A'\). Therefore, set A must be \(A = \{4, 6, 8, 9\}\).

Step 4 ：For part (b), we are given that set B is \(B = \{3, 5, 7, 8, 9\}\). To find the complement of set B, denoted as \(B'\), we need to find all elements in the universal set U that are not in B.

Step 5 ：By comparing the elements of the universal set U with the elements of B, we find that the elements 4 and 6 are not in B. Therefore, \(B'\) must be \(B' = \{4, 6\}\).

Step 6 ：Final Answer: For part (a), \(\boxed{A = \{4, 6, 8, 9\}}\). For part (b), \(\boxed{B' = \{4, 6\}}\).