1. Let X be the Boolean hypercube {0,1}n. For a set I ⊆ {1,2, . . . ,n} we define a parity function hI as follows. On a binary vector x = (x1, x2, . . ., xn) ∈ {0,1}n , hI (x) = _ _ i∈I xi _ mod 2 . (That is, hI computes parity of bits in I .) What is the VC-dimension of the class of
all such parity functions, Hn-parity = {hI : I ⊆ {1,2, . . .,n}}?
2 We proved Sauer’s lemma by proving that for every class H of finite VC-dimension d, and every subset A of the domain, |HA| ≤ |{B ⊆ A : H shatters B}| ≤ _d i=0 _|A| I _ . Show that there are cases in which the previous two inequalities are strict (namely, the ≤ can be replaced by ) and cases in which they can be replaced by equalities. Demonstrate all four combinations of = and .
