Minimum closure of F (functional dependencies) help

• KataKoniK
In summary, the minimum closure of F can be obtained by eliminating redundant attributes, combining FDs with the same left-hand side, and removing transitive dependencies. The resulting closure is {S -> NOPQZVWX, NX -> U, NXT -> R, YQ -> N, YQ -> X}.
KataKoniK

Homework Statement

F = {S -> NOPQZVWX, NX -> UV, NXT -> RY, YQ -> NXR, QZ -> S, PQ -> S}

Find the minimum closure of F.

none

The Attempt at a Solution

So far I have made RHS of each FD into a single attribute and then I eliminated redundant attributes from LHS.

This is what I got. Is it correct so far? Do I still have to delete more stuff? I don't know what to do with QZ -> S and PQ -> S in terms of if they should be removed or not.

{NX -> U, NX -> V, S -> N, S -> O, S -> P, S -> Q, S -> Z, S -> V, S -> W, S -> X, NXT -> Y, QZ -> S, PQ -> S, YQ -> N, YQ -> X}

Your approach is a good start, but there are a few more steps you can take to minimize the closure of F. Here are some suggestions:

1. Eliminate redundant attributes from the right-hand side of the functional dependencies. For example, in the FD NX -> UV, the attribute V is redundant since it is already implied by U. So the FD can be simplified to NX -> U.

2. Combine FDs with the same left-hand side. In this case, the FDs S -> NOPQZVWX and S -> V can be combined into one FD, S -> NOPQZVWXV.

3. Remove transitive dependencies. For example, in the FD NXT -> RY, the attribute Y is transitively dependent on NXT since it can be determined from the FD YQ -> NXR. So the FD can be simplified to NXT -> R.

After applying these steps, the minimum closure of F would be:

{S -> NOPQZVWX, NX -> U, NXT -> R, YQ -> N, YQ -> X}

Note that the FDs QZ -> S and PQ -> S are not redundant and should not be removed, as they cannot be implied by any other FD in the closure.

I would suggest using the Armstrong's axioms and closure algorithm to find the minimum closure of F. This method systematically eliminates redundant functional dependencies and ensures that the resulting set is a closure of F. In this case, the minimum closure of F would be:

{S -> NOPQZVWX, NX -> UV, NXT -> RY, YQ -> NXR}.

This is because all the other dependencies can be derived from these four using the Armstrong's axioms. Additionally, QZ -> S and PQ -> S cannot be removed as they are not redundant and are necessary for the closure of F.

1. What is the minimum closure of F (functional dependencies) and why is it important?

The minimum closure of F, also known as the canonical cover, is the smallest set of functional dependencies that can generate all the other functional dependencies in a given set. It is important because it helps in simplifying a large set of functional dependencies, making it easier to understand and work with.

2. How do you calculate the minimum closure of F?

To calculate the minimum closure of F, you need to follow a three-step process:

1. Determine all the trivial functional dependencies, which are of the form A → A, where A is any attribute in the set.

2. Eliminate any redundant functional dependencies by checking if they can be derived from the other functional dependencies in the set.

3. Remove any extraneous attributes from the functional dependencies, which are attributes that do not contribute to the left-hand side or right-hand side of the dependency.

3. What is the difference between the minimum closure and the closure of F?

The minimum closure is the smallest set of functional dependencies that can generate all the other functional dependencies in a given set, while the closure of F is the set of all functional dependencies that can be derived from the given set of functional dependencies. The minimum closure is a simplified version of the closure of F.

4. Can the minimum closure of F be used to determine all the candidate keys in a relation?

Yes, the minimum closure of F can be used to determine all the candidate keys in a relation. Candidate keys are those attributes or combination of attributes that can uniquely identify each tuple in a relation. The minimum closure of F helps in identifying the minimal set of attributes that can determine all the other attributes in a relation, which are the candidate keys.

5. How does the minimum closure of F help in database design?

The minimum closure of F helps in database design by simplifying a large set of functional dependencies, making it easier to identify the primary key and candidate keys for a relation. It also helps in reducing data redundancy and improving data integrity by ensuring that each attribute in a relation is only dependent on the primary key or candidate keys.

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