I have to find a really bad example where you cannot apply Heaths Theorem because of a missing functional dependency.

But am I right, that every relation has at least one functional dependency?

Or can anyone of you come up with an example table with at least 3 Sets of Attributes to project it and then join it together. The result should of course be different from the original table to proof Heath's Theorem right (no FD -> no lossless decompositions)

It does not matter, wheter it is an sql example or an abstract one.

  • I know this is no relevant question for programmers, because such an example would never be considered practically relevant. But the proof alone is not enough for my Prof. :(
    – BackfromHell
    Dec 3, 2013 at 12:30

1 Answer 1


But am I right, that every relation has at least one functional dependency?

Relation can has no functional dependencies, if it has no non-key attributes, i.e. it is full-keyed relation.

  • 1
    You should strike through that incorrect sentence where you say "yes, I believe". Then what remains is the correct answer. Jan 3, 2014 at 22:46
  • @Erwin Smout, I have done.
    – msi77
    Jan 4, 2014 at 6:42
  • This answer is wrong. Always 1 or more FDs hold. Version 1 is correct, except the "because" leaves out the case of no attributes, since then there is still a FD. I don't know what @Erwin was thinking (re version 2) since they know the answer. Maybe they were thinking of non-trivial FDs, not FDs. Also the situation is confusing because the question post doesn't ask exactly 1 question & the title suggests a question opposite to the quoted question from the body. So it's a good idea to not say "yes" or "no" in the answer but just state what is so--as you do here, although your answer is wrong.
    – philipxy
    Apr 28, 2020 at 17:10
  • There are always FDs. However there are not always non-trivial FDs & there are not always FDs other than the ones implied by the CKs. Moreover, even if this were clearly talking about such CKs, it's still not clear what exactly you are trying to say via the "if". All attributes prime is neither necessary nor sufficient for no CKs except the ones implied by the CKs.
    – philipxy
    Apr 28, 2020 at 17:26

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