Consider the relation R$($A$,$B$,$C$,$D$,$E$,$F$)$ with the following set of functional dependen$-$ cies. Assume there are no set$-$valued attributes.

A$->$B

B $->$ CD E$->$F D$->$A

$($a$)$ Show that AE qualifies as a key for R$.$

$($b$)$ In the rest of the question, assume AE is the key of R$.$ Find all violations of $1$NF$,2$NF$,3$NF$,$ and BCNF $($if any$).$ Clearly mark which violation is for which normal form.

$($c$)$ Overall, which normal form is R in$?$ If R is not in any normal form, write $$none$.$

$($d$)$ Does R satisfy $2$NF$?$ If not, decompose it to achieve $2$NF$.$

$($e$)$ Does your result from part d satisfy $3$NF$?$ If not, decompose it to achieve $3$NF$.$

$($f$)$ Does your result from part e satisfy BCNF$?$ If not, decompose it to achieve BCNF$.$

$($g$)$ Consider the relations obtained at the end of part f$.$ Is this set of relations a dependency$-$

preserving decomposition of R according to the functional dependencies given in the question? Why or why not?

$($h$)$ Consider the relations obtained at the end of part f$.$ Is this set of relations a lossless decomposition of R$?$ Why or why not?