Functional dependencies proving method

dependenciesrelations

I was having some difficulties when trying to use Armstrong axiom to solve functional dependencies.

R (A,B,C,D,E,F)
Given FD = { A->CD , C->BE, B->F, F->A }
Prove A-> BF

By using closure, for A, I get {A,C,D,B,E,F} so I know A->BF is valid.

So I came out with these to prove using Axiom:

A->CD
CD->BED (Augmentation)
A->BED (Transitivity)
BED->FED (Augmentation)
A->FED (Transitivity)

Then from there, I am lost. Anybody knows how to prove it?

Thanks in advance.

Best Answer

Here is a possible derivation:

 1. A → CD   (given)
 2. C → BE   (given)
 3. CD → BED (by augmentation of 2 with D)
 4. A → BED  (by transitivity of 1 and 3)
 5. B → F    (given)
 6. B → BF   (by augmentation of 5 with B)
 7. BED → B  (by reflexivity)
 8. A → B    (by transitivity of 4 and 7)
 9. A → BF   (by transitivity of 8 and 6)
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