1. **State the problem:** We are given a partially ordered set (poset) $(S, \leq)$ with $S = \{1, 2, 4, 8, 9, 10, 15, 16\}$ and a subset $A = \{8, 9\}$. We need to find the maximal element, minimal element, maximum element, minimum element, upper bound, lower bound, least upper bound (supremum), and greatest lower bound (infimum) of $A$. 2. **Understand the order relation:** The problem doesn't explicitly state the order relation, but usually for subsets of numbers with $\leq$, the order is the usual numeric order. We assume $\leq$ is the standard order on numbers. 3. **Maximal and minimal elements of $A$:** - The **maximal element(s)** of $A$ are elements of $A$ such that no other element in $A$ is strictly greater. - Since $A = \{8, 9\}$, both elements are incomparable only to themselves. Since $9 > 8$, $9$ is the maximal and also maximum element. - The **minimal element(s)** of $A$ are elements with no other element in $A$ strictly smaller. - Since $8 < 9$, $8$ is minimal and also minimum element. 4. **Maximum and minimum elements:** - The **maximum** element is the unique greatest element in $A$. Here, $9$ is the maximum. - The **minimum** element is the unique least element in $A$. Here, $8$ is the minimum. 5. **Upper bounds of $A$ in $S$:** - An **upper bound** of $A$ is an element $u \in S$ such that $x \leq u$ for all $x \in A$. - Since $A = \{8,9\}$, $u$ must satisfy $8 \leq u$ and $9 \leq u$. - Candidates in $S$ satisfying this are $9, 10, 15, 16$. 6. **Lower bounds of $A$ in $S$:** - A **lower bound** of $A$ is an element $l \in S$ such that $l \leq x$ for all $x \in A$. - $l$ must satisfy $l \leq 8$ and $l \leq 9$. - Candidates in $S$ satisfying this are $1, 2, 4$ because all are less than or equal to both 8 and 9. 7. **Least upper bound (supremum) of $A$ in $S$:** - The **least upper bound** is the smallest element in $S$ among all upper bounds of $A$. - Upper bounds are $\{9, 10, 15, 16\}$, and the smallest of these is $9$. 8. **Greatest lower bound (infimum) of $A$ in $S$:** - The **greatest lower bound** is the largest element in $S$ among all lower bounds of $A$. - Lower bounds are $\{1, 2, 4\}$, and the largest of these is $4$. **Final answers:** - Maximal element of $A$: $9$ - Minimal element of $A$: $8$ - Maximum element of $A$: $9$ - Minimum element of $A$: $8$ - Upper bounds of $A$: $\{9, 10, 15, 16\}$ - Lower bounds of $A$: $\{1, 2, 4\}$ - Least upper bound (supremum) of $A$: $9$ - Greatest lower bound (infimum) of $A$: $4$