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 \(s^{\prime}\) 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 " \(s\) ' \(\Rightarrow\) \(\left(l-s^{\prime}ight)\) " cannot be more than the confidence of " \(s \Rightarrow(l-s)\)," where \(s^{\prime}\) 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 \(D\).