Partial dependency

Question

The number of partial dependencies in a relation R{X,Y,V,U,Z} with FD set F={X->YZ, Z->V, V->XY} is ____

Comments

A  functional dependency X → Y is a  partial dependency if some attribute  A ε X can be removed from  X and the dependency still holds; that is, for some  A ε X,(X – {A}) → Y.

Nowhere in the question it is asked about the partial dependencies On CANDIDATE KEYS, so why the answers are based on this assumption..??

I think there are NO functional dependencies because you cannot remove any attribute from LHS..!

Answer 1

Candidate keys are XU,VU,ZU. So prime attributes= X,U,Z,V and non prime attributes= Y.

Partial dependency means that a non prime attribute is functionally dependent on part of a candidate key. (A non prime attribute is an attribute that's not part of any candidate key.)

In the dependencies non prime attribute Y is dependent on X which is part of candidate key XU.  So X->YZ is a partial dependency.

Similarly Y is dependent on V which is part of candidate key VU.

So,there exists 2 partial dependencies which are X->YZ and V->XY.

Comments

X->YZ and V->XY themselves are not partial dependencies, but X->Y and V->Y (which can be implied from F) are partial dependencies. Hence, we can say that the set F has 2 partial dependencies.

---

z--->y can also be derive from the above given FDs.. and z is the proper subset of zu and y is a non-prime attribute.. so I think it is also  a partial dependency.

Answer 2

In the above question there are 2 partial dependencies X->YZ and V->XY because Y is a non prime attribute which is depending on a part of candidate keys( UX,UV,UZ )

Answer 3

Candidate keys are:

XU,ZU,VU

Prime attributes={X,U,Z,V}

So,partial dependency is in:X->YZ,V->XY

and there exists no dependency in Z->V because they both are prime attributes.

Answer 4

Candidate Keys: UX , UV and UZ

Prime Attributes: { U, V, X, Z}

All the three FD are partial dependencies. (directly or indirectly after RHS split)

As a proper subset of Candidate keys gives non-prime attributes.

Answer 5

So, total partial dependencies=2

Answer 6

The dependencies are:

         X$\rightarrow$YZ,  Z$\rightarrow$V, V$\rightarrow$XY

By the rule of decomposition, we can write:

X$\rightarrow$Y, X$\rightarrow$Z , Z$\rightarrow$V, V$\rightarrow$X, V$\rightarrow$Y

And, XU, VU and ZU are the candidate keys, so prime attributes are: {X,V,U,Z}

and as such X$\rightarrow$Y and V$\rightarrow$Y are partial dependencies.