  Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set
system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{
\triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at
most $(2^{d\bmod{k}}+o(1))\binom{n}{\lfloor d/k\rfloor}$. Here $\triangle$
denotes the symmetric difference operator. This is a $k$-fold generalisation of
a result of Dvir and Moran, and it settles one of their questions. A key
insight is that, by a compression method, the problem is equivalent to an
extremal set theoretic problem on $k$-wise intersection or union that was
originally due to Erd\H{o}s and Frankl. We also give an example of a family
$\mathcal{A} \subset 2^{[n]}$ such that the VC dimension of $\mathcal{A}\cap
\mathcal{A}$ and of $\mathcal{A}\cup \mathcal{A}$ are both at most $d$, while
$\lvert \mathcal{A} \rvert = \Omega(n^d)$. This provides a negative answer to
another question of Dvir and Moran.
