First test losslessness.
$R_1\cap R_2=\{A\}$.
We are given
$A\to DE$.
By reflexivity, $A\to A$.
$\therefore A\to ADE$.
But $ADE=R_2$.
Hence, $(R_1\cap R_2)\to R_2$.
Therefore, the decomposition is lossless.
So A is correct.
Now determine the important projected dependencies.
On $R_1(A,B,C)$
From the original FDs $:C\to A$ holds directly.
Also, $A\to D$ and $D\to B$, so $A\to B$.
Thus useful dependencies on $R_1$ include $C\to A$ and $A\to B$.
On $R_2(A,D,E)$
We directly have
$A\to DE$.
Also, from $DE:D\to B$,
then $BE\to C$,
then $C\to A$.
$\therefore DE\to A$ also belongs to the projection.
Now ask whether all original dependencies can be recovered from the union of these projected FD sets.
Consider $D\to B$.
Starting with $D$ using the projected FDs, we cannot derive $B$.
Thus, $D\to B$ is not preserved.
So C is correct.
Now consider $BE\to C$.
Starting from $BE$ using only the projected dependencies, we cannot obtain the attributes needed to derive $C$.
$\therefore BE\to C$ is also not preserved.
So D is correct.
Since at least one original FD cannot be inferred from the projected FD sets, the decomposition is not dependency preserving.
Thus B is incorrect.
