1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

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

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 *H*1*, . . .,**Hr *be hypothesis classes over some fixed domain set *X*. Let *d *= max*i *VCdim(*Hi *) and assume for simplicity that *d *≥ 3.

3. Prove that VCdim _ ∪*ri* =1 *Hi* _ ≤ 4*d *log(2*d*)+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 2*k *possible labelings on these examples. Use Sauer’s lemma to show that the union class cannot produce more than *rkd *labelings. Therefore, 2*k **rkd *. Now use Lemma A.2.