$3.$ In the following transaction database, if the minimum support is $7,$ please list all frequent itemsets and their support counts. $[10$ points$]$

TID Transaction

$100\{$A$,$ B$,$ E$\}$

$200\{$B$,$ D$\}$

$300\{$A$,$ B$,$ E$\}$

$400\{$A$,$ C$\}$

$500\{$B$,$ C$\}$

$600\{$A$,$ C$\}$

$700\{$A$,$ B$\}$

$800\{$A$,$ B$,$ C$,$ E$\}$

$900\{$A$,$ B$,$ C$\}$

$1000\{$A$,$ C$,$ E$\}$

$4.$ In Question $3$ above, please answer the following questions:

$($a$)$ draw the lattice of all itemsets $($with their support counts$)[10$ points$]$;

$($b$)$ Given minimal support $5,$ list closed and maximal frequent itemset$($s$)[10$ points$].$