I have the following problem:
Let $X=\{2, 3, 4, 5, \ldots\}$ be ordered by division, that is:
$$x\leq y\Leftrightarrow x\mid y,$$ and let $\mathcal{S}=\{(A, \leq_A): A\subset X\}$ where $\leq_A$ is the order induced on $A$ by $\leq$ via restriction. Then we can order $\mathcal{S}$ via inclusion, that is:
$$A\leq_{\mathcal{S}} B\Leftrightarrow A\subset B.$$
Find the minimal and maximal elements of $\mathcal{S}$ via $\leq_{\mathcal{S}}$.
What I don't understand about this problem is: why should one consider $\mathcal{S}$ as subsets of $X$ ordered by the induced order? I mean, why can't it just be: Let $\mathcal{S}$ be the subsets of $X$ ordered by inclusion?
The minimal elements should be the unitary sets and the maximal element should be $X$, right?