I have some difficulties with understanding the notion of VC dimension.
The following is the number of question I want to answer.
Q: If there is a set $|S|=k$ such that hypothesis space $H$ doesn't shatter it, does it mean that $VC(H)<k$?
Yes. According to the definition if H doesn't shatter the set of size k, it doesn't shatter any possible set of size k, however here we have some set $S$, so there are might other set $S'$ of size $k$ which is shatter by $H$.
Q:if $H_1$ and $H_2$ are two hypothesis spaces and $H_1 \subseteq H_2$ then $VC(H_1) \leq VC(H_2)$?
Yes. Every hypothesis that belongs to $H_1$ are in $H_2$ so the shattered subset of $H_2$ is at least as of $H_1$.
Did I answer it correctly?