Given the set of functional dependencies:

AB-C
AD-E
B-D
AF-B

Does AB-E?

Question

Given the set of functional dependencies:

AB->C
AD->E
B->D
AF->B

Does AB->E?

BlackTom AI · Accepted Answer

Reformulating the problem, we’re asked whether the functional dependency AB -> E is implied by the given set of dependencies: AB -> C, AD -> E, B -> D, AF -> B.

First, I’ll compute the closure of AB (denoted AB+), since a dependency AB -> E is implied exactly when E is in AB+. Starting with AB, we can apply AB -> C to add C to AB+. Now AB+ = {A, B, C}.

Next, the rule B -> D lets us add D to AB+. So AB+ = {A, B, C, D}.

We also have AD -> E. Since AB+ already contains A and D, we can apply AD -> E to add E to AB+. Therefore AB+ = {A, B, C, D, E}.

At this point, E is present in AB+, which means AB -> E is satisfied by the augmentation of AB with the rest of the attributes in the closure. In other words, within the given dependency set, E is functionally determined by AB.

A note on the other dependency AF -> B: this one isn’t needed for deriving AB -> E, because we already have a path from AB to E through AB -> C and B -> D followed by AD -> E. There isn’t any contradiction or scenario that would negate AB -> E using the provided FDs.

In summary, AB+ contains E, confirming that AB -> E is implied by the given functional dependencies. The reasoning shows the crucial steps: from AB we gain C, from B we gain D, and with A and D we gain E via AD -> E, completing the chain to AB -> E.

HADM3740/HADM6740 Relational Model and Functional Dependencies Quiz

Given the set of functional dependencies: AB->C AD->E B->D AF->B Does AB->E?

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