$2.$ BCNF Consider a relation schema R with attributes ABCDEFGH with functional dependencies S:

S $=\{$FG $->$ CDEF, EFH $->$ AB$,$ CDH $->$ ADG, CEH $->$ FH$,$ pBD $->$ EG$,$ F $->$ BCG$\}$

$($a$)$ Which of these functional dependencies violate BCNF$?$

$($b$)$ What are all the candidate $($minimal$)$ keys of R$?$

$($c$)$ Employ the BCNF decomposition algorithm to obtain a lossless decomposition of R into a collection of relations that are in BCNF$.$ Make sure it is clear which relations are in the final decomposition and project the dependencies onto each relation in that final decomposition.

$($d$)$ Is your decomposition dependency$-$preserving? If not state at least one dependency that is lost.

Explain all your answers and show your work for partial credit.