$(1)$ A database has five transactions $($T$100$ to T$500)$ as shown in the table below. Let min$\_$sup $=3$

and min$\_$conf $=80\%.$

a$.$ Find all the frequent itemsets using the Apriori algorithm. You must show the contents of $C_k$

and $L_k$ tables in each step $($please refer to your lecture slides for examples$).$ Hint: the items

must be in alphabetical order.

b$.$ List all the association rules with a confidence measure. Indicate which ones are strong rules.

$(2)$ The following incomplete contingency table summarizes supermarket transaction data, where

hot dogs refer to the transactions containing hot dogs, hot dogs refer to the transactions that do

not contain hot dogs, hamburgers refer to the transactions containing hamburgers, and

$$ hamburgers refer to the transactions that do not contain hamburgers.

a$.$ Complete the contingency table.

b$.$ Suppose that the association rule "hot dogs $=>$ hamburgers" is being mined. Given a

minimum support threshold of $25\%$ and a minimum confidence threshold of $50\%,$ is this

association rule strong? Show the calculation to support your answer.

c$.$ Based on the given data, is the purchase of hot dogs independent of the purchase of

hamburgers? If not, what kind of correlation relationship exists between the two? Show the

Lift calculation to support your answer.