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I am seeking for help. I am stuck over a question based on functional dependency.

Consider the relation schema R= (A, B, C, D, E, G) and the set F of functional dependencies: A → BC BD → E CD → AB a. Derive the nontrivial dependencies from F using Armstrong’s axioms: AD → E and CD → E. Show all steps.

I can not understand the question. As per my understanding, here all functional dependencies are non trivial. So what is the meaning of Armstrong’s axioms: AD → E and CD → E.

Also, what is the meaning of derivation of nontrivial dependencies?

philipxy
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Encipher
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1 Answers1

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Let’s try to derive AD → E from the Armstrong’s axioms (I think this is the request of the exercise).

Here are the steps of derivation:

1. A → BC (given)
2. BD → E (given)
3. AD → BCD (for augmentation from 1)
4. BCD → BD (for reflexivity)
5. AD → BD (for transitivity from 3 and 4)
6. AD → E (for transitivity from 5 and 2)

I left the derivation of CD → E as exercise.

Renzo
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  • Thank you so much sir, So derivation means given the FD, A → BC BD → E CD → AB I need to find out how could I reach to AD->E and CD->E? Am I right? It helps me to learn from your derivation. – Encipher Nov 19 '21 at 18:51
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    Yes, the exercise asked how to derive both AD->E and CD->E using the Armstrong’s axioms. – Renzo Nov 19 '21 at 19:18
  • How can I get BCD->BD? Which functional dependency help to get this? – Encipher Nov 19 '21 at 20:24
  • For BCD->BD means BD is the subset of BCD. Which functional dependency implies that? – Encipher Nov 19 '21 at 20:52
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    The [reflexivity axiom](https://en.wikipedia.org/wiki/Armstrong%27s_axioms#Axiom_of_reflexivity) says: if Y is a subset of X, the X -> Y holds. Here BD is a subset of BCD. – Renzo Nov 19 '21 at 21:22
  • For the second one CD->AB(given), CD ->A, CD->B (Decomposition) CDD -> AD [Augmentation], CD->AD, AD->E then CD ->E [Transitivity]. Is it correct? – Encipher Nov 19 '21 at 23:49
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    Yes, it is correct. – Renzo Nov 20 '21 at 13:19