DM: Functional Dependency (weak form)

[Michael Walther]

=======================
Hello,

I have a question regarding functional dependency. I have a table 
with 
say 4 attributes, A, B, C, D. Now after processing through the table I
have found that the following functional dependencies hold good on 
the 
table:

       A->B, A->D, B->C, D->A, D->B, D->C.

 Now my question is that from this set of functional dependencies can
 I say that the following also holds good:

  A->BD, D->ABC   -  (Is this also true)
 and is
  AC->BD true.

 Thanks for the answer, Regards
 -Srikanth
                                                
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If A->B denotes the following :
There is a relation R between A and B
(i.e. a subset from the cross product AB from A and B)
with :
For every a from A there is a non empty subset S(a) from AB (*).
S(a) := {(a,b)| a R b with b from B}
(A and B are weak functional dependent)

You can infer from A->B, that A->BD holds.
A->B yields S(a)  for every a from A with relation R and
A->D yields S'(a) for every a from A with relation R'.
For every a from A we have S''(a) :=
{(a,(b,d))| a R b with b from B and a R' d with d from D }
so we have also a relation R'' between A and BD with (*) 
hence A-> BD.
The same works for D->ABC if D->A, D->B, D->C is true.
For a relation AC-> BD you can construct 
S((a,c)):={((a,c),(b,d))| a R b with b from B and a R' d with d from 
D}
for every given (a,c) from AC. Hence AC-> BD.

This works also for strong functional dependency, i.e. A->B denotes
the existence of a function.
 regards
 Michael Walther
 mail : 100721,15
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