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Relation R(ABCDE) with FD set $F=\left \{ A\rightarrow BC,C\rightarrow DE,D\rightarrow E \right \}$

and decomposition are $R1\left ( ABCD \right )$ and $R2\left ( DE \right )$

Now my question is $C\rightarrow DE$ dependency preserved? They told it is preserved, because

$C\rightarrow D$ and $D\rightarrow E$ these two dependencies preserving and that is why $C\rightarrow DE$ also preserving. Is it true?

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C -> D is preserved in the first relation, D -> E in the second one.

Now when we want to check if C -> DE is preserved, we consider the union of the FDs obtained from the decomposed relations and then try whether we are able to get what we need or not. So in this case, we have C -> D and D -> E with us.

When we take closure of C, we get C -> CDE, and thus, we are able to get C - > DE in this decomposition.
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U mean transitive dependency?We have to check transitive dependency to check some dependency valid or not? Can u give me some resource?
There is no dependency directly from $C\rightarrow E$, is it not a violation of dependency?