1. **State the problem:** We are given the total number of elements in the universal set $\xi$ as $n(\xi) = 21$, the number of elements in the union of sets $A$ and $B$ as $n(A \cup B) = 19$, the number of elements in $A$ but not in $B$ as $n(A \cap B') = 8$, and the total number of elements in set $A$ as $n(A) = 12$. We need to complete the Venn diagram with these values. 2. **Recall formulas and rules:** - The union formula: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ - The complement of $B$ in $A$ is $A \cap B'$, so $$n(A) = n(A \cap B) + n(A \cap B')$$ - The universal set $\xi$ contains all elements, so $$n(\xi) = n(A \cup B) + n((A \cup B)')$$ 3. **Find $n(A \cap B)$:** Given $n(A) = 12$ and $n(A \cap B') = 8$, then $$n(A \cap B) = n(A) - n(A \cap B') = 12 - 8 = 4$$ 4. **Find $n(B)$:** Using the union formula: $$19 = 12 + n(B) - 4$$ $$n(B) = 19 - 12 + 4 = 11$$ 5. **Find $n(B \cap A')$:** Since $n(B) = n(B \cap A) + n(B \cap A')$, and $n(B \cap A) = n(A \cap B) = 4$, then $$n(B \cap A') = n(B) - n(A \cap B) = 11 - 4 = 7$$ 6. **Find elements outside both $A$ and $B$:** $$n((A \cup B)') = n(\xi) - n(A \cup B) = 21 - 19 = 2$$ 7. **Summary for the Venn diagram:** - Outside both $A$ and $B$: 2 - Only in $A$: 8 - In both $A$ and $B$: 4 - Only in $B$: 7 **Final answer:** $$\boxed{\text{Outside } A \text{ and } B = 2, \quad A \text{ only} = 8, \quad A \cap B = 4, \quad B \text{ only} = 7}$$