1. **Stating the problem:** We need to find the maximal and minimal elements of the poset (partially ordered set) defined on the set $\{3,5,9,15,24,45\}$. 2. **Understanding the poset:** Usually, for sets of numbers, the poset is defined by the divisibility relation: $a \leq b$ if and only if $a$ divides $b$. 3. **Minimal elements:** An element is minimal if no other element in the set divides it except itself. 4. **Maximal elements:** An element is maximal if it does not divide any other element in the set except itself. 5. **Check divisibility among elements:** - 3 divides 9, 15, 24, 45 - 5 divides 15, 45 - 9 divides 45 - 15 divides none except itself - 24 divides none except itself - 45 divides none except itself 6. **Identify minimal elements:** - 3 is divisible by no other element except itself - 5 is divisible by no other element except itself So minimal elements are $\{3,5\}$. 7. **Identify maximal elements:** - 15 is not divided by any other element except itself - 24 is not divided by any other element except itself - 45 is not divided by any other element except itself So maximal elements are $\{15,24,45\}$. **Final answer:** - Minimal elements: $\{3,5\}$ - Maximal elements: $\{15,24,45\}$