Please answer #$4$ and $5.$

Problem $4$: Consider the following relation:

$R(A,B,C,D,E)$

And the following set of functional dependencies $F$ :

$A,CE$

$ED$

$BC,E$

$BD$

$($a$)$ Using the FDs$,$ determine the key K of R$.$ Hint: there is only one possible key for R$.$

$($b$)$ Determine a minimal basis $B$ for $F.$

$($c$)$ Using $K$ and $B$ determined above, calculate the $3NF$ of this database using the synthesis

algorithm. Show the resulting relations and their keys.

Problem $5.$ We now want to convert $R$ from Problem $4$ to $BCNF$ with respect to the minimal

basis $B$ using the splitting algorithm. The algorithm would begin as follows:

$S=\{R\}$

$R$ is not in $BCNF$ with respect to $A,CE.$ Therefore, we split it into:

$R1(A_{,}{C}_{,}D,E)$

$R2(B_{,}{B}_{,}C)$

$S=\{R1,R2\}$

Note that the key of $R2$ carries over from the key of the relation that was split $($R$).$

Continue the splitting algorithm until all relations are in BCNF$.$ Show your work. Hint: you should

need to do two more splits.