Assume that the relational $R$ with schema $R(A,B,C,D,E,F),$ with the following set $(F)$ of functional dependencies. $F=\{ABC$;$CD$;$DEF\}.$ If $R$ is decomposed into three relations$-$

$R1(A,B,E),R2(A,B,C,F),R3(A,B,D),$ do the following.

$($a$)$ Check if the above decomposition is lossless or not.

Use matrix approach discussed in the lectures$/$textbook$.$ You are required to give the initial matrix, mention the possible modifications to the matrix $,$ and give the final answer$).$

$($b$)$ Give the projection of $F$ on $R1,R2$ and $R3($i$.$e$.,{}_{R1}(F),pro{d}_{R2}(F)$ and $\{$:${}_{R3}(F)).$