$($a$)(2\%)$ Define the general definition of $3$NF$,$ 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 $3$NF$.$

$($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.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.