$(35$ points$)$ Given a relation $R(A,B,C,D,E,F,G,H)$ and a set $T$ of functional

dependencies, $T=\{ABEF,CDA,DG,$ABD$$EFG,$GH\}$:

$3$a$)$ Draw all functional dependencies on $R$ using the diagram introduced in class.

$3$b$)$ List ALL candidate keys of $R.$

$3$c$)$ Determine whether each of the following dependencies are implied by T$?$

A$.AF($YES$)($NO$)$

B$.DFGH,($YES$)($NO$)$

C$.$ CD $$ E $($YES$)($NO$)$

D$.$ ABF$GH($YES$)($NO$)$

E$.$ AD $$ F $($YES$)($NO$)$

$3$d$)$ Find a minimal cover for $T.$

$3$e$)$ What is the highest normal form of R$?$ Can it be normalized further? Briefly

explain.

Consider the decomposition into $2$ relations: $($ABCDEF$)($DGH$).$ What are the

normal forms the decomposition satisfies? Is it a lossless$-$join decomposition?

Is it a dependency preserving decomposition? Explain your answers.

$3$g$)$ Consider the decomposition into $4$ relations: $($ACD$)($ABEF$)($DG$)($GH$).$ What

are the normal forms this decomposition satisfies? Is it a lossless$-$join

decomposition? Is it a dependency preserving decomposition? Explain your

answers.