As exercise 2.32 shows that the powerset preorder is a symmetric monoidal preorder, it suffices to check whether it has all joins.

Let \$$A\subseteq\mathbb{P}(S)\$$. \$$\vee A=\cup A\$$. The existence of \$$\cup A\$$ is guaranteed by the [Axiom of Union](https://en.wikipedia.org/wiki/Axiom_of_union). So, the only thing to check is that \$$\cup A\subseteq S\$$. Let \$$x\in\cup A\$$. Then (by the Axiom of Union) there is a set \$$D\in A\$$ such that \$$x\in D\$$. For every \$$D'\in A\$$, \$$D'\subseteq S\$$, so in particular, \$$D\subseteq S\$$. Thus, \$$x\in D\Rightarrow x\in S\$$ (since \$$x\$$ was arbitrary, this holds for every \$$x\in\cup A\$$). So, \$$\cup A\subseteq S\$$, as was needed to prove that the powerset symmetric monoidal preorder is a quantale.