Consider the following set F of functional dependencies on the relation schema R$($A$,$ B$,$ C$,$ D$,$ E$,$ F$)$:

A$->$ D

AB$->$ DE

BD$->$ A

AC$->$ F

$($a$)$

List two candidate keys for R$.(6$ points$)$

$($b$)$

Give a lossless dependency preserving $3$NF decomposition of r based on the set of functional dependencies. $(12$ points$)$

$($c$)$

Give a lossless BCNF decomposition of r based on the set of functional dependencies. Does the decomposition preserve dependencies? Why or why not?