Question $2.(50$ marks$)$

Consider the relation R $=\{$A$,$ B$,$ C$,$ D$,$ E$,$ F$,$ G$,$ H$,$ I, J$\}$ and the set of functional dependencies F $=\{\{$A$,$ B$\}->\{$C$\},\{$A$\}->\{$D$,$ E$\},\{$B$\}->\{$F$\},\{$F$\}->\{$G$,$ H$\},\{$D$\}->\{$I$,$ J$\}\}.$

$($a$)$ Proof $\{$A$\}->\{$E$,$I$\}$ holds by using inference rules. $(10$ marks$)$

$($b$)$ Whether $\{$A$,$B$,$D$\}$ is a super key? Whether $\{$A$,$B$,$D$\}$ is a candidate key? Why? $(10$ marks$)$

$($c$)$ Whether $\{$A$,$F$\}$ is a super key? Whether $\{$A$,$F$\}$ is a candidate key? Why? $(10$ marks$)$

$($d$)$ Decompose R into $2$NF but not $3$NF$.(10$ marks$)$

$($e$)$ Decompose R into $3$NF based on the results of $($d$).(10$ marks$)$