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As it says in the title I have trouble understanding why if we have X->A and Y->B then why is it wrong to write XY->AB. They way I understand it, if A is functionally dependent of X and B is functionally dependent of Y, then when we have XY on the left side we should have their corresponding values on the right side. Anyway my book says that this is wrong, so can anyone give me an example where this is proven wrong ? Thanks in advance :)

Mike Sherrill 'Cat Recall'
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Dimitar Spasovski
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2 Answers2

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You're going about this the wrong way.

In order for "{X->A, Y->B}, therefore XY->AB" to be true, you need to prove that you can derive XY->AB from {X->A, Y->B}, using only Armstrong's axioms and the additional rules derived from Armstrong's axioms.

Mike Sherrill 'Cat Recall'
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  • So you're saying that although it seems fine when you look at in logical way, it can be proven wrong using those axioms, so it is wrong. Thanks a lot, that really changed the way I think about this type of problems. – Dimitar Spasovski May 04 '14 at 15:42
  • No, I'm saying that "{X->A, Y->B}, therefore XY->AB" can't be proven *true* from those axioms. (There may well be a way to prove that's false, but I can't do it off the top of my head.) – Mike Sherrill 'Cat Recall' May 04 '14 at 17:48
  • @MikeSherrill'CatRecall' can you please have a look at my answer below ?Since you have such a high reputation it would be hard that you are wrong. – PleaseHelp Jan 02 '15 at 10:02
  • Oh, I can be wrong. Especially in formal logic, which isn't my field. – Mike Sherrill 'Cat Recall' Jan 03 '15 at 15:40
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    @MikeSherrill'CatRecall': Proof: `1) X -> A` `2) Y -> B` `3) XY -> AY {Augmentation 1 & Y` `4) XY -> A {Decomposition 3` `5) XY -> XB {Augmentation 2 & X` `6) XY -> B {Decomposition 5` `7) XY -> AB {Union 4 & 6` – Pragy Agarwal May 20 '15 at 23:25
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If X uniquely determines A and similarly Y uniquely determines B ,then any combination of XY uniquely determines AB.

Hence , X->A ,Y->B infers XY->AB is true.

More supporting links.

http://en.wikipedia.org/wiki/Functional_dependency…

See the composition rule here. Not crebile enough ? Then in the following link , Slide 9 says that

Textbook, page 341: ”… X A, and Y B does not imply that XY AB.” Prove that this statement is wrong.

http://www.ida.liu.se/~TDDD37/fo/fo-normalization

Moreover, Mike's answer is trying to prove the "vice versa" , which may not necessarily be true.

PleaseHelp
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  • The OP needs to see how to *prove* this claim using Armstrong's axioms (reflexivity, augmentation, transitivity). – Mike Sherrill 'Cat Recall' Jan 03 '15 at 15:44
  • I am unable to see where has the OP mentioned "Armstrong's axioms" . Moreover, I think you tried to prove the "vice versa" , which may not necessarily be true. And the statement is actually true. – PleaseHelp Jan 03 '15 at 15:59
  • Armstrong's axioms are sound and complete rules of inference with respect to functional dependencies. Any FD that holds can be derived from those three axioms. (I was, indeed, answering a question the OP didn't ask. I've cut that from my answer.) – Mike Sherrill 'Cat Recall' Jan 03 '15 at 17:53