$(6$ marks$)$ Consider the relational schema R$($A$,$ B$,$ C$,$ D$,$ E$,$ G$,$ H$,$ I, J$)$ and a set of functional dependencies F $=\{$A $->$ EH$,$ BE $->$ G$,$ DI $->$ AJ$,$ GH $->$ CI$,$ C $->$ BDI$\}.$ Note that A$,$ B$,$ C$,$ D$,$ E$,$ G$,$ H$,$ I and J are attributes. Regarding F$,$ given a decomposition R$1=\{$ADIG$\},$ R$2=\{$ABEG$\},$ R$3=\{$CEGH$\},$ and R$4=\{$BECJ$\}$ of R$.$ i$.$ Is R$1$ in $3$NF$?$ Please justify your answer. $(3$ marks$)$ ii$.$ Is the decomposition lossless? Please justify your answer. $(3$ marks$)($b$)(14$ marks$)$ Consider the relational schema R$($A$,$ B$,$ C$,$ D$,$ E$,$ G$,$ H$,$ I, J$)$ and the set of functional dependencies F $=\{$AE $->$ CGH$,$ BE $->$ IJG, D $->$ AI$,$ GJ $->$ DE$,$ AHI $->$ B$\}.$ Note that A$,$ B$,$ C$,$ D$,$ E$,$ G$,$ H$,$ I and J are attributes. Justify your answer to each question. i$.$ Determine the highest normal form of R$.(2$ marks$)$ ii$.$ Find all the candidate keys for R$.(2$ marks$)$ iii. Find a minimal cover Fm for F$.(3$ marks$)$ iv$.$ Decompose R into BCNF$.(4$ marks$)$ v$.$ From your answer in iv$.,$ is your decomposition dependency preserving? $(3$ marks$)$