Consider the following transactions:

TID

Transaction

T$1$

apple, banana, pear, fig

T$2$

banana, orange, fig

T$3$

fig, water, pear, orange, milk

T$4$

pear, banana, water, fig

T$5$

pear, water, fig

T$6$

kiwi, pear, milk, water, fig

T$7$

apple, pear, orange, water, banana, kiwi

T$8$

pear, orange, apple

Using the min support of $50\%$ and the confidence of $70\%$:

$($A$)$ Find all frequent itemsets using the Apriori algorithm. Show how the Apriori algorithm

works in a step by step manner.

$($B$)$ List all the association rules found by the Apriori algorithm.

$($C$)$ Find the Frequent Patterns using FP$-$Tree algorithm.