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I was wondering that if a certain hypothesis class $H$ has a VC dimension $d$ over domain $X$ how to prove that $H$ will shatter all subsets of $X$ with size less than $d$ i.e $H$ will shatter $A \subset X$ where $|A| \leq d-1$.

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We can show that it is not true by a counterexample. For example, $X = \{1,2,3\}$ and $\mathcal H = \{\{\},\{1\},\{2\},\{1,2\}\}$ is the finite set hypothesis class. By the definition, $\mathcal V \mathcal C$ dimension of $\mathcal H$ over domain $X$ is $d=2$. Although, $A = \{3\} \subset X$, whose size is smaller than the $\mathcal V \mathcal C$ dimenion i.e $|A|<d=2$ it is not shattered by $\mathcal H$.

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