Question: Find all connected and path-connected subsets of $\mathbb{Q}$
We've shown that the connected subsets of $\mathbb{Q}$ are all sets that contain exactly one element of $\mathbb{Q}$.
Now our reasoning is as follows: Since path-connected subsets of $\mathbb{Q}$ are connected as well, all path-connected subsets of $\mathbb{Q}$ are only the single-element subsets as well.
Is this reasoning correct?