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Attribute set closures

0

$begingroup$

Consider the following relationship with its attributes:
$$R( V, W, X, Y, Z )$$

This relationship has the following functional dependencies:
$$FD: { XY rightarrow W; V rightarrow Y; Z rightarrow X }$$

From this, I need to find all elements in the attribute set closure ${VW}^+$

I am having trouble figuring out the solution to this problem. From what I gather, some transformations can be made:

$$V rightarrow Y Rightarrow XV rightarrow XY$$
$$Z rightarrow X Rightarrow XZ rightarrow XY$$

An then to figure out the set items, I am trying to use an induction method which states that if the left side of a functional dependency is a subset of the closure, then the right side of the dependency is a part of the closure's elements.

$$ V rightarrow Y: V subset {VW}^+ therefore Y in {VW}^+ $$

However, this is the only element I can seem to find using this induction method. $W$ is not on the left side of any dependencies I have found, and none of the transformations I have found are a subset of the closure.

The solution states that the following are examples of what should be included:

$${V}$$
$${VW}$$
$${Y}$$
$${VWY}$$

I cannot figure out how these elements are included. Can someone please explain?

relation-algebra

share|cite|improve this question

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

$endgroup$

add a comment | 

0

$begingroup$

Consider the following relationship with its attributes:
$$R( V, W, X, Y, Z )$$

This relationship has the following functional dependencies:
$$FD: { XY rightarrow W; V rightarrow Y; Z rightarrow X }$$

From this, I need to find all elements in the attribute set closure ${VW}^+$

I am having trouble figuring out the solution to this problem. From what I gather, some transformations can be made:

$$V rightarrow Y Rightarrow XV rightarrow XY$$
$$Z rightarrow X Rightarrow XZ rightarrow XY$$

An then to figure out the set items, I am trying to use an induction method which states that if the left side of a functional dependency is a subset of the closure, then the right side of the dependency is a part of the closure's elements.

$$ V rightarrow Y: V subset {VW}^+ therefore Y in {VW}^+ $$

However, this is the only element I can seem to find using this induction method. $W$ is not on the left side of any dependencies I have found, and none of the transformations I have found are a subset of the closure.

The solution states that the following are examples of what should be included:

$${V}$$
$${VW}$$
$${Y}$$
$${VWY}$$

I cannot figure out how these elements are included. Can someone please explain?

relation-algebra

share|cite|improve this question

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

$endgroup$

add a comment | 

$begingroup$

Consider the following relationship with its attributes:
$$R( V, W, X, Y, Z )$$

This relationship has the following functional dependencies:
$$FD: { XY rightarrow W; V rightarrow Y; Z rightarrow X }$$

From this, I need to find all elements in the attribute set closure ${VW}^+$

I am having trouble figuring out the solution to this problem. From what I gather, some transformations can be made:

$$V rightarrow Y Rightarrow XV rightarrow XY$$
$$Z rightarrow X Rightarrow XZ rightarrow XY$$

An then to figure out the set items, I am trying to use an induction method which states that if the left side of a functional dependency is a subset of the closure, then the right side of the dependency is a part of the closure's elements.

$$ V rightarrow Y: V subset {VW}^+ therefore Y in {VW}^+ $$

However, this is the only element I can seem to find using this induction method. $W$ is not on the left side of any dependencies I have found, and none of the transformations I have found are a subset of the closure.

The solution states that the following are examples of what should be included:

$${V}$$
$${VW}$$
$${Y}$$
$${VWY}$$

I cannot figure out how these elements are included. Can someone please explain?

relation-algebra

share|cite|improve this question

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

$endgroup$

share|cite|improve this question

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

share|cite|improve this question

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

edited Mar 24 at 2:03

Andrés E. Caicedo

66.1k8160252

asked Mar 23 at 23:02

HausHaus

1114

asked Mar 23 at 23:02

HausHaus

1114