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

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

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

of functions and some function are called Dudley classes.

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