I have a relation R(A,C,D) where AC is the key and the FD set for the relation is F = {AC -> D, D -> C}.
I need to convert this 3NF relation to BCNF. However the result of decomposition is R(A,C,D) (exactly the same relation) and R(D,C), where AC and D are keys respectively.
Can a component relation of a decomposition be the same as the original relation?
If one of the components of a decomposition is the same as the original and the original is not in the desired NF (normal form) then the component is not in the NF & the decomposition isn't a decomposition to the desired NF.
If one of the components of a decomposition is the same as the original and the original was in the desired NF then you didn't need to decompose.
Proven algorithms for decompositions to higher NFs always return components that are smaller than the original or return just the original. Otherwise they wouldn't be useful since all they would have done is add relations when you already had just one.