Let $C\subseteq 2^X$ be a concept class over $X$ and let $\bar{C}:=\{X\setminus c\mid c\in C\}$ be the complement. Show that $VCdim(C)=VCdim(\bar{C})$.
Proof:
Let $d:=VC_{dim}(C)$, then there exists $S\subseteq X$, $|S|=d$, s.t. $S$ is shattered by $C$.
Let $d':=VC_{dim}(\bar{C})$, then there exists $S'\subseteq X$, $|S'|=d'$, s.t. $S'$ is shattered by $C$.
Show that $d\leq d'$ and $d' \leq d$. I know that a set $S$ is shattered by $C$ iff $\Pi_C(S):=\{c\cap S\mid c\in C\}=2^S$, but I have no clue how to show the two sides. Can someone help me with that?