1. **State the problem:** We have a universal set $U = \{1, 2, \ldots, 20\}$, and three subsets: - $X = \{4, 5, 6, 7, 8\}$ - $P = $ prime numbers in $U = \{2, 3, 5, 7, 11, 13, 17, 19\}$ - $O = $ odd numbers in $U = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ We need to identify which Venn diagram correctly represents these sets and their intersections. 2. **Check each option for correctness:** - **Option a:** Circle $O$ contains $9$ (correct, since $9$ is odd). Circle $X$ contains $4,5,6,7,8$ (correct). Circle $P$ contains $2,3,5,7,11,13,17,19$ (correct). Outside circles: $10,12,14,16,18,20$ (correct, these are even and not prime or in $X$). This matches all definitions. - **Option b:** Circle $O$ contains $3,7,9,11,13,15,17,19$ but $9$ is outside and only $3$ inside $O$ (contradiction). $9$ must be inside $O$ because it is odd. - **Option c:** Circle $O$ contains $1,3,7,9,11,13,15,17,19$ excluding $9$ (contradiction, $9$ is odd and must be inside $O$). - **Option d:** Circle $O$ contains $1,3,5,7,9,11,13,15,17,19$ including $5$ and $9$. $5$ is prime and in $X$, so it should be in the intersection of $O$, $P$, and $X$ if applicable. But $5$ is odd, so it belongs to $O$. This is consistent. However, $5$ is in $X$ and $P$, so it should be in the intersection area of all three circles. This option does not specify overlaps clearly but includes $5$ in $O$ which is correct. 3. **Conclusion:** Option a correctly places all elements in their respective sets and intersections, including $9$ inside $O$, and the prime numbers correctly in $P$. Therefore, the correct diagram is option a. **Final answer:** Option a represents the situation correctly.