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Say we have a relation R(A, B, C). We have one known functional dependency: A → B. I'm wondering if it's true that there would be no way to prove that A → B implies B → A. If it was true, I can't think of a proof with Armstrong's axioms. Is there a name for this? I believe asymmetry and antisymmetry state something stronger.

A → B says that any tuple sharing the same values of the attribute A must have the same values of the attribute B.

Thus this relation is valid:

A B C
1 1 1
1 1 2
2 2 1

But then so is this one:

A B C
1 1 1
1 1 2
2 1 1

Because the tuple (2, 1, 1) does not violate A → B.

Is this correct?

  • Yes, it's correct. A proof that A → B does not imply B → A is your counterexample. – ypercubeᵀᴹ Nov 26 '18 at 22:54

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