The Apriori algorithm makes use of prior knowledge of subset support properties.
(a) Prove that all nonempty subsets of a frequent itemset must also be frequent.
(b) Prove that the support of any nonempty subset s 0 of itemset s must be at least as great as the support of s
(c) Given frequent itemset l and subset s of l, prove that the confidence of the rule "s' ⇒ (l - s')" cannot be more than the confidence of "s ⇒ (l - s)," where s 0 is a subset of s.
(d) A partitioning variation of Apriori subdivides the transactions of a database D into n nonoverlapping partitions. Prove that any itemset that is frequent in D must be frequent in at least one partition of D.