Let $n$ be a positive integer.

Let $S \subseteq \mathbb{R}^n$. Is the Hausdorff dimension of the boundary of $S$ always smaller than the Hausdorff dimension of $S$?

I have not found anything concerning those questions in some looked up books, I was not able to prove one of the statements, and I failed finding a counterexample. Does anybody know something about that?