1. (*) Prove that for r = 2 it holds that VCdim(H1 ∪H2) ≤ 2d +1.

2. Dudley classes: In this question we discuss an algebraic framework for defining concept classes over Rn and show a connection between the VC dimension of such classes and their algebraic properties. Given a function f : Rn →R we define the corresponding function, POS( f )(x) = 1[ f (x)>0]. For a class F of real valued functions we define a corresponding class of functions POS(F) = {POS( f ) : f ∈ F}. We say that a family, F, of real valued functions is linearly closed if for all f , g ∈ F and r ∈R, ( f +rg)∈F (where addition and scalarmultiplication of functions are defined point wise, namely, for all x ∈ Rn, ( f +rg)(x)= f (x)+rg(x)). Note that if a family of functions is linearly closed then we can view it as a vector space over the reals. For a function g : Rn →R and a family of functions F, let F + g d=ef { f + g : f ∈ F}. Hypothesis classes that have a representation as POS(F + g) for some vector space

of functions F and some function g are called Dudley classes.