How is #14 false?

[bbell25cmc]

Can someone explain to me why question 14 in the quiz practice questions for wednesday is false? If MhN = 2^N, shouldn't H shatter X?

[mikeizbicki]

There is a subtle difference between "there exists" and "for all" in the definitions that this problem is getting at.

In particular, $m_{\mathcal H}=2^N$ implies that there exists at least one dataset $X$ of size $N$ that can be shattered. This does not imply that every dataset of size $N$ will be shattered, and so we cannot guarantee that $X$ will be shattered. It is in fact common to have hypothesis classes that can only shatter some datasets of a particular size.

[bbell25cmc]

But since N is the N of that specific dataset, shouldn't it shatter that specific dataset?