Lossless Join Property and One Example

Question

We know: Lossless-Join Decomposition on http://en.wikipedia.org/wiki/Lossless-Join_Decomposition

If we have Relation R(A,B,C,D) and Functional Dependencies F={A->B, B->C, C->D} , so the

R1(A,D), R2(B,D), R3(C,D)

is not loos-less decomposition , but R1(A,B), R2(A,C), R3(A,D) is loos-less decomposition .

anyone could describe me as simple as possible why this is true?

Answer 1

R1(A,D), R2(B,D), R3(C,D)

hasn't the special property mentioned in your wiki links,

but

R1(A,B), R2(A,C), R3(A,D) 

has the special property in which that the intersection is A and A in X F+ for all attributes X .