Question B$.[50$ marks$]$

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

$1)$ Prove that $\{$A$\}\{$G$,$J$\}$ holds using inference rules. $[10$ marks$]$

$2)$ Identify the candidate key and explain the reasoning behind your choice. $[10$ marks$]$

$3)$ Determine if $\{$A$,$C$,$D$\}$ is a superkey? Why or why not? $[10$ marks$]$

$4)$ Decompose R into $2$NF but not $3$NF$.[10$ marks$]$

$5)$ Decompose the relation R into $3$NF$.[10$ marks$]$