Given , set of dependecies X --> Y , Y --> Z , Z --> W and W --> Y and the table is decomposed into (x,y) , (y,z) and (y,w).
Lossless property is satisfied since intersection of (x,y) , (y,z) is y and y+ = yz which is superkey for (y,z) hence we can merge them .Then intersection (x,y,z) and (y,w) is y+ = yw since Y --> Z and Z --> W.Hence the given decomposition is lossless.
Now coming to dependency preservation , we have to write all the direct and implied dependencies for each of the subrelations.Lers write for each of the subrelation one by one.
a) For (x,y) : X --> Y(direct)
b) For (y,z) : Y --> Z(direct) , Z --> Y(implied from Z --> W and W --> Y)
c) For (w,y) : W --> Y(direct) , Y --> W(implied from Y --> Z and Z --> W)
Now for checking the dependency preservation , we have to check from the original FD set whether each of the FDs are satisfied from the FDs which are there in the subrelations.
Now X --> Y , Y --> Z and W --> Y are also there in subrelations so no concern here.
For Z --> W , we have found the implied dependencies Z --> Y and Y --> W from which we can obtain Z --> W.Hence this FD is also satisfied in the subrelation.Hence the given decomposition is dependency preservation also.
Hence C) is the correct option.
