Checking for lossless-join property

Question

The decomposition of relation R with FD set F into R1 and R2 has lossless join property iff
R1 $\cap$ R2 $\rightarrow$ R1 $\in$ F$^+$
  OR
R1 $\cap$ R2 $\rightarrow$ R2 $\in$ F$^+$

If R is decomposed into more than 2 relations, can we apply this same rule pairwise to all possible pairs to check for lossless join property ? Or is that table drawing thing the only way to check for lossless join property ?

Answers with reference(s) will be much appreciated. Thanks.

Comments

Yes wt u told is correct about losess join .we can apply safely in more than two relation.!

Answer 1

Let us take R , S , T be decomposed subrelations of A.So to check lossless property , we do:

a) Let us start by taking  R ∩ S.If common attribute exist in this intersection and that is the superkey of either R or S or both , then we take natural join of R and S and then proceed.We take now intersection of T and join of R and S and hence find common attribute.If that is not superkey of T or R join S , then we have to start with other pair and repeat the same procedure.

Else we can terminate and can conclude that it is a lossless join decomposition.For the 1st case , that is failure of finding superkey , we have to start with other pair ,e.g. we start with intersection of S and T.

b) If in the above case ,where we have started taking intersection of R and S,suppose we get a common set of attribute which is not a superkey for either R or S,then we stop right there and start with other pair , e.g. start taking the common attribute of S and T first and so on.

So in total we have to consider 3 cases here :

i) start with R,S

ii) start with S,T

iii) start with R,T

If we fail on getting common attribute at last following the procedure described above which is superkey , then we can conclude that the decomposition is lossy.

I hope I have addressed your doubt.