$1.$ Consider a relation R$($E$,$ B$,$ V$,$ D$)$ with the set of functional dependencies F $=\{$EB $->$ V$,$ BV $->$ D$,$ VD $->$ E$\}.$ Assume that R is decomposed into R$1($E$,$ B$,$ V$)$ and R$2($E$,$ V$,$ D$).$ Find whether the given decomposition is lossless or not.

The following decomposition is lossless. The decomposition is lossless because the R$=$ R$.$

$2.$ Consider a relation R $($A$,$ W$,$ C$,$ O$)$ and a set of functional dependency $\{$AW $>$ C$,$ C $>$ O$,$ O $>$ A$\}.$ Relation R is decomposed into R$1($A$,$ W$,$ C$)$ and R$2($C$,$ O$).$ Check whether the given decomposition is dependency preserving or not.Problem $520$ pts

Consider a relation $R(E,B,V,D)$ with the set of functional dependencies

$V,BVD,VDE.$ Assume that $R$ is decomposed into $R_1(E,B,V)$ and $,$

$V,D.$ Find whether the given decomposition is lossless or not.

The following decomposition is lossless. The decomposition is lossless because

the $R^{\text{'}}=R.$

Consider a relation $R(A,W,C,O)$ and a set of functional dependency $,$

$CO,OA.$ Relation $R$ is decomposed into $R1(A,W,C)$ and $R2(C,O).$ Check

whether the given decomposition is dependency preserving or not.