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The following questions examine how the support and confidence of an association rule may vary in the presence of a concept hierarchy.

(a) Consider an item x in a given concept hierarchy. Let x1, x2, …, xk denote the k children of x in the concept hierarchy. Show that s(x) ≤ k i=1 s(xi), where s(·) is the support of an item. Under what conditions will the inequality become an equality?

(b) Let p and q denote a pair of items, while ˆp and ˆq are their corresponding parents in the concept hierarchy. If s({p, q}) > minsup, which of the following itemsets are guaranteed to be frequent? (i) s({p, q ˆ }), (ii) s({p, qˆ}), and (iii) s({p, ˆ qˆ}).

(c) Consider the association rule {p} −→ {q}. Suppose the confidence of the rule exceeds minconf. Which of the following rules are guaranteed to have confidence higher than minconf? (i) {p} −→ {qˆ}, (ii) {pˆ} −→ {q}, and (iii) {pˆ} −→ {qˆ}.