Is my decomposition of R into 3NF is lossless and preserving all dependencies? I have a schema R = {A,B,C,D,E} and the set F of FDs:

F = {A --> BCDE, CD --> E, EC --> B}.

Candidate key = A
Non - Prime Attributes = BCDE
Prime Attributes = A

According to me, there is no partial dependency because no non-prime attribute is dependent on part of CK.

E and B are transitively dependent on A so the schema Ris not in 3NF.

Decomposing R into 3NF, I have three relations and three set of FDs : 

R1(A,B,C,D)                         F1={A --&gt; B,A --&gt; C,A --&gt; D}
R2(C,D,E)                           F2={CD --&gt; E}
R3(E,C,B)                           F3={EC --&gt; B}


My question is, Is my decomposition of R into 3NF is lossless and preserving all dependencies or not? And if not how can I decompose it into 3NF losslessly while preserving all dependencies.
So My decomposition is lossless and preserving all dependencies but there is no need for relation R3(E, C, B) because we already have A which can derive B in R1.