I'm having some trouble with Problem 2.13 -- especially Part C. Let me walk through the other parts first...

For Part A, if the VC dimension of a hypothesis set is

, then you need at least

hypotheses to run the

points through in order to get them to shatter. So,

.

For Part B, at worst, your hypothesis set contains no, or perhaps 1, hypothesis. In this event, you have

. I claim you can't have a VC dimension of more than

. Assume to the contrary that you could shatter

points in the intersection; hence, you can find

points that are shattered by

. But since these hypotheses belong to every individual

, it means that you can shatter these points in the

that has the minimum dimension -- a contradiction. So,

.

For Part C, it seems obvious that your lower bound is always the maximum VC dimension from the individual sets:

-- simply because the hypotheses that shatter

points in the corresponding set will carry through to the union.

The upper bound is harder. I can think of simple cases (i.e., 2 hypothesis spaces, each with 2 hypotheses in it) where

. While looking ahead a little at Problem 2.14, I'm trying to see if I can find a case where

. I can't find one, so I'm just hoping that

is a break point. I want to argue as follows: present me with

points that are shattered, and it must be the case that either

or

shattered

or

points, which is a contradiction. So, given the points, if you restrict to the subset of the union that contains all hypotheses in

, by assumption it can't shatter more than

points, so some dichotomy is missing...but then I get stuck.

Any hints?