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.

1. Let be the Boolean hypercube {0,1}n. For a set ⊆ {1,2, . . . ,n} we define a parity function hI as follows. On a binary vector x = (x1x2, . . ., xn) ∈ {0,1}hI (x) = _ _ iI xi _ mod 2 . (That is, hI computes parity of bits in .) What is the VC-dimension of the class of

all such parity functions, Hn-parity = {hI ⊆ {1,2, . . .,n}}?

2 We proved Sauer’s lemma by proving that for every class of finite VC-dimension d, and every subset of the domain, |HA| ≤ |{⊆ shatters B}| ≤ _d i=0 _|AI _ . 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 .

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Suggest a modification of the binary search algorithm that emulates this strategy for a list of names.

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Draw a class diagram that shows the relationships among the classes in this new version of the system

Jack decides to rework the banking system, which already includes the classes BankView, Bank, SavingsAccount, and RestrictedSavingsAccount. He wants to add another class for checking accounts. He sees that savings….