Q$4.$

$($a$)(2\%)$ Define the general definition of $3NF,$ which considers all keys of a relation.

$($b$)(6\%)$ Derive the undesirable dependencies when a functional dependency $x$A violates $3$NF$.$ You need to clearly explain how the undesirable dependencies are derived from $x$ A violating the conditions of $3NF.$

$($c$)(4\%)$ How does Boyce$-$Codd Normal Form $($BCNF$)$ differ from $3$NF$?$ Explain why $3$NF adopts a weaker condition than BCNF does.

$($d$)(2\%)x$ is a key in a relation $R.$ Prove that $x$ A for any attribute $A$ in $R,$ according to the definition of functional dependency.