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

dependencies F $=\{\{$A$,$ D$\}->\{$C$\},\{$A$\}->\{$B$,$ E$\},\{$D$\}->\{$F$\},\{$F$\}->\{$G$,$ H$\},\{$B$\}->\{$I$,$ J$\}\}.$

$1)$ Prove that $\{$A$\}->\{$B$,$ J$\}$ holds by using inference rules. $(10$ marks$)$

$2)$ Is $\{$A$,$ B$,$ D$\}$ a superkey? Is $\{$A$,$ D$,$ E$\}$ a candidate key? Why? $(10$ marks$)$

$3)$ Is $\{$C$,$ D$\}$ a superkey? Is $\{$C$,$ D$\}$ a candidate key? Why? $(10$ marks$)$

$4)$ Decompose R into $2$NF$.(10$ marks$)$

$5)$ Decompose R into BCNF$.(10$ marks$)$