2. 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 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 (ls) cannot be more than the confidence ofs(ls), where s 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.