Can an independent set (of vertices) be a vertex cover as well?
If something satisfies the definition of&nbsp;$X$ and the definition of&nbsp;$Y$, then it is both an&nbsp;$X$ and a&nbsp;$Y$.

Although it's true that, whenever $S$&nbsp;is independent, $V\setminus S$ is a vertex cover, it's possible for both $S$ and $V\setminus S$ to be independent: this happens when $S$&nbsp;is one side of a bipartite graph, as is the case in the example in the question.