Consider a market basket transactions shown in the following Table $3.$

Table $3.$ Market basket transactions

$($a$)$ How many different association rules $xY($where $x$ and $Y$ are non$-$empty itemsets$)$ can be extracted

from this data $($including rules that have zero support$)?$ Write down the calculation details.

$($b$)$ Assume that the given minimum support is $0\mathrm{.}1.$ What is the maximum size of frequent itemsets that

can be extracted?

$($c$)$ How many different $3-$itemsets $f($i$.$e$.,3-$sized itemsets $f,|f=3)$ that can be derived from the above

transactions? Write down the calculation details.

$($d$)$ Find an itemset $f$ of size $2$ or larger $($i$.$e$.,|f|2)$ that has the largest support.

$($e$)$ Find two $1-$itemsets a and $b$ with $|a|=1$ and $|b|=1,$ such that the rules $\{a\}\{b\}$ and $\{b\}\{a\}$ have

the same confidence.

$($f$)$ Assume that the given minimum support is $0\mathrm{.}3,$ apply the priori algorithm to find all frequent itemsets

in the above transactions. Write down the calculation details.