I got stuck with this problem:
Given just a $$X=\{1,2,3,4\}$$ and a $$C=\{\{1,3,4\},\{4\},\{2,3\},\{3\},\{1,2\},\{1\},\{2\}\}$$
how do I calculate VC dimension of C?
I know that for example with $$S=\{1,2,3\}$$ given in the prior problem would easily lead me to the end. This because $$S \subseteq X$$ So I would just have to verify if $$|S\cap C|$$ (known as the cardinality of the intersection) can be decomposed in a "2 power of D". So given $$|S \cap C| = 8$$ since $$8 = 2^3$$ I will have $$D=3$$ and if $$|S| = D \to VCdim(C)\ge D$$
This because by definition
$$VCdim(C)=max\{D|\exists S, |S|=D, |\pi_c(S)=2^D\}$$ $$|\pi_c(S)|=\{c_i\cap S| c_i\in C\}$$ where $$|\pi_c(S)|$$ is the cardinality of the intersection.
The problem that I'm facing is: if the S is not given, how can I find a legit S, which by definition I know it is the biggest set that can be shattered from C?