1. Prove that for every H that is PAC learnable, VCdim(H)∞. (Note that this is the implication 3→6 in Theorem 6.7.)

2.  VC of union: Let H1, . . .,Hr be hypothesis classes over some fixed domain set X. Let d = maxi VCdim(Hi ) and assume for simplicity that d ≥ 3.

3. Prove that VCdim _ ∪ri =1 Hi _ ≤ 4d log(2d)+2log(r ).

Hint: Take a set of k examples and assume that they are shattered by the union class. Therefore, the union class can produce all 2k possible labelings on these examples. Use Sauer’s lemma to show that the union class cannot produce more than rkd labelings. Therefore, 2k rkd . Now use Lemma A.2.