1. **State the problem:** We are given sets $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{2, 4, 6, 8\}$, and $B = \{1, 2, 3, 5, 7\}$. We need to find which option equals $A' \cup B$, where $A'$ is the complement of $A$ relative to $U$. 2. **Recall the complement definition:** The complement $A'$ consists of all elements in $U$ that are not in $A$. So, $$A' = U \setminus A = \{1, 2, 3, 4, 5, 6, 7, 8\} \setminus \{2, 4, 6, 8\} = \{1, 3, 5, 7\}$$ 3. **Calculate the union $A' \cup B$:** $$A' \cup B = \{1, 3, 5, 7\} \cup \{1, 2, 3, 5, 7\} = \{1, 2, 3, 5, 7\}$$ 4. **Compare with given options:** - Option A: $\emptyset$ (empty set) — no - Option B: $\{2\}$ — no - Option C: $\{4, 6, 8\}$ — no - Option D: $B = \{1, 2, 3, 5, 7\}$ — yes - Option E: None of these — no **Final answer:** $A' \cup B = B$, so the correct choice is **D**.