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.

{U -> VW, W -> R} |= {U -> VWR}

Explanation / Answer

Given- {U->VW,W->R}

To Prove- {U->VW, W->R}={U->VWR}.

We need to prove {U->VW, W->R} derives {U->VWR} and {U->VWR} derives {U->VW, W->R}.

To check- {U->VWR} derives from {U->VW, W->R}.

U->VW can be written as U->V, U->W.( Using Decomposition Rule)

U->W and W->R implies U->R (Using Transitive Rule)

Now we have U->W, U->V, U->R implies U->VWR.(Using Union Rule)

Hence we can derive functional dependency {U->VWR} derives from {U->VW,W->R}.

To check- {U->VWR} derives {U->VW, W->R}.

U->VWR can be written as U->V, U->W, U->R.

So we have U->V and U->W implies U->VW (Using Union Rule).

But we cannot get W->R.

So {U->VWR} does not derives {U->VW, W->R}.

Hence , It is proved that {U ->VW, W->R} ={U->VWR} is wrong.

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