Decomposition:

A decomposition of R into R1,R2,...,Rn is valid if R1,..,Rn make up all the attributes in R
and is given by:

R1 = project_{attributes of R1} (R)
R2 = project_{attributes of R2} (R)
...
Rn = project_{attributes of Rn} (R)


A good decomposition is:

     1. lossless: required property [all decompositions should be lossless!]
     2. dependency preserving [desired property!]

Lossless
--------
A decomposition of R into R1,R2,..Rn is lossless iff we are guaranteed for every
possible instance of R that  R = R1*R2*...*Rn


Projection of functional dependencies to a decomposition
--------------------------------------------------------

Given a relation R and a set F of fd, the projection of F into a decomposed
relation R1 is F1 and F1 is given by the set of all functional dependencies in F+
that only contain the attributes in R1.

R(A,B,C,D)   F = {A->B, B->C, C->D}

Ex.
R1(A,B,C)    F1 = {A->B, B->C}
R2(C,D)      F2 = {C->D}

Ex2.
R1(A,B,D)    F1 = {A->B, A->D, B->D}    
R2(C,D)      F2 = {C->D}

"Algorithm" for finding projection: Find all subsets X of attributes of R1,
compute closure wrt F, add X -> X+ into F1. Find the minimal basis.


Functional dependency preserving decomposition
-----------------------------------------------

A decomposition of R and a set of fds F into R1,R2,..Rn is functional
dependency preserving if, say F1,...,Fn are projection of F into R1,...,Rn
then:

F is equivalent to F1 union F2 union ... union Fn.
(meaning that they have the same closure!)


Check for fd preservation:

R(A,B,C,D)   F = {A->B, B->C, C->D}

Ex.
R1(A,B,C)    F1 = {A->B, B->C}
R2(C,D)      F2 = {C->D}


F1 union F2 = {A->B, B->C, C->D}, this is equal to F.
This is fd preserving.



Ex2.
R1(A,B,D)    F1 = {A->B, A->D, B->D}    
R2(C,D)      F2 = {C->D}

F1 union F2 = {A->B, A->D, B->D, C->D}
F = {A->B, B->C, C->D}

(Since F1 union F2 is obtained by projection, then everything
in F1 and F2 are implied by F)

Check if F1 union F2 implies everything in F:

A->B:    in F1 union F2
B->C:    compute wrt F1 union F2:  B+ = {B,D}, C is not implied. 
C->D:    in F1 union F2

B->C is not preserved. So this is not a dependency preserving decomposition.