$($a$)$ A database has $9$ transactions, shown below:

TID Items Bought

T$1\{$A$,$ B$\}$

T$2\{$B$,$ D$\}$

T$3\{$B$,$ C$\}$

T$4\{$B$,$ A$,$ D$\}$

T$5\{$A$,$ C$\}$

T$6\{$B$,$ C$\}$

T$7\{$A$,$ C$\}$

T$8\{$B$,$ A$,$ C$,$ E$\}$

T$9\{$B$,$ A$,$ C$\}$

Assuming a minimum level of support count min sup count $=2$ and a minimum level of confidence min conf $=50\%$ :

$($i$)$ Use the Apriori algorithm to find the frequent itemset$($s$).$ Then list all frequent itemsets that can be derived from the frequent itemset$($s$)$ of the algorithm $($not just the ones with the maximum width$/$length$).$ Just showing the final answer is not acceptable; show all steps of the algorithm with appropriate justification. For each iteration $($k $)$ show the candidate $($Ck $)$ and acceptable frequent itemsets $($Lk $).$

$($ii$)$ List all of the association rules that can be derived based on the frequent itemset$($s$)$ of the previous question. While showing the association rules, show also their support and confidence values. Finally, mention and explain which of these association rules are strong.