1. **Problem statement:** Given sets with operations and elements: - $A \cup B = \{a, c, e, 3, 4, 6, 7\}$ - $A \cap B = \{c, 6\}$ - $A \setminus B = \{a, e, 4\}$ Find $B$. 2. **Recall set operation definitions:** - $A \cup B$ is the union of $A$ and $B$, all elements in either set. - $A \cap B$ is the intersection, elements common to both. - $A \setminus B$ is the difference, elements in $A$ but not in $B$. 3. **Find $A$ first:** Since $A \setminus B = \{a, e, 4\}$ and $A \cap B = \{c, 6\}$, then $$A = (A \setminus B) \cup (A \cap B) = \{a, e, 4\} \cup \{c, 6\} = \{a, c, e, 4, 6\}$$ 4. **Find $B$ using union and intersection:** We know $$A \cup B = \{a, c, e, 3, 4, 6, 7\}$$ and $$A = \{a, c, e, 4, 6\}$$ Since union includes all elements in $A$ or $B$, elements in $B$ but not in $A$ are $$B \setminus A = (A \cup B) \setminus A = \{3, 7\}$$ 5. **Use intersection to find $B$:** $$B = (B \setminus A) \cup (A \cap B) = \{3, 7\} \cup \{c, 6\} = \{3, 7, c, 6\}$$ **Final answer for part (a):** $$B = \{3, 7, c, 6\}$$