Prove or disprove the following proposed inference rules for functional dependen

Question

Prove or disprove the following proposed inference rules for functional dependencies. A proof should be made by using the reflexive, augmentation, transitive, decomposition, union, and pseudotransitive rules. A disproof should be performed by demonstrating a relation instance that satisfies the conditions and functional dependencies on the left hand side of the proposed rule but does not satisfy the dependencies on the right hand side. No other form of dispoof will be accepted.

{UV -> WR, S -> TV, T -> R} |= {US -> W}

Explanation / Answer

{UV -> WR, S -> TV, T -> R} |= {US -> W}
Here the set is F(RSTUVW)
Given,
UV -> WR , S -> TV, T -> R
Then
U cannot individually determine anything in this set,
U -> U (Reflexive Rule as U is subset of U)
S -> STV(Reflexive Rule as S is subset of S and given that S -> TV. So by Composition Rule)
US -> USTV(Composition Rule)

So,US -> W is not satisfied
Hence disproved.

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