Dependency preserving decomposition - solved exercises 3

Dependency preserving decomposition - Dependency preserving decomposition solved exercises - How to verify that a decomposition is dependency preserving? - Steps to find dependency preserving decomposition - Dependency preserving decomposition examples

Dependency preserving decomposition

Consider a relation R (A, B, C) with the following set of functional dependencies;

F = {A → B, B → C}.

Is the decomposition of R (A, B, C) into R1 (A, C) and R2 (B, C) a dependency preserving decomposition?

Solution:

The above said decomposition of R into R1 and R2 is a dependency preserving decomposition if (F₁ U F₂)⁺ = F⁺, where F₁ is set of FDs hold by R1, F₂ is set of FDs hold by R2, and F is the set of FDs hold by R. (F₁ U F₂)⁺ is the closure of (F₁ U F₂), and F⁺ is the closure of F.

Step 1: for R1, the non-trivial functional dependency is A → C. Hence F1 = {A → C}.

Step 2: for R2, the non-trivial functional dependency is B → C. Hence, F2 = {B → C}.

Step 3: find F1 U F2 and the closure of F1 U F2.

F1 U F2 = {A → C, B → C} ≠ F.

(F1 U F2)⁺ = {A → C, B → C}

F⁺ = {A → B, B → C, A → C}

And, (F1 U F2)⁺ ≠ F.

The FD A → B is not supported by (F₁ U F₂)⁺. Hence, the decomposition of R (A, B, C) into R1 (A, C) and R2 (B, C) is not a dependency preserving decomposition.

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Dependency preserving decomposition - solved exercises 2

Dependency preserving decomposition - Dependency preserving decomposition solved exercises - How to verify that a decomposition is dependency preserving? - Steps to find dependency preserving decomposition - Dependency preserving decomposition examples

Dependency preserving decomposition

Consider a relation R (A, B, C, D) with the following set of functional dependencies;

F = {A → B, B → C, and C → D}.

Is the decomposition of R (A, B, C, D) into R1 (A, C, D) and R2 (B, C) a dependency preserving decomposition?

Solution:

The above said decomposition of R into R1 and R2 is a dependency preserving decomposition if (F₁ U F₂)⁺ = F⁺, where F₁ is set of FDs hold by R1, F₂ is set of FDs hold by R2, and F is the set of FDs hold by R. (F₁ U F₂)⁺ is the closure of (F₁ U F₂), and F⁺ is the closure of F.

Step 1: for R1, the derivable non-trivial functional dependency is, C → D. Hence, F₁ = {C → D}

Step 2: for R2, the derivable non-trivial functional dependency is, B → C. Hence, F₂ = {B → C}

(F₁ U F₂) = ({C → D} U {B → C}) = {C → D, B → C} ≠ F.

(F₁ U F₂)⁺ = {C → D, B → C, B → D}

F⁺ = {A → B, B → C, C → D, A → C, A → D, B → D}

Hence, (F₁ U F₂)⁺ ≠ F⁺

The FD A → B is not supported by (F₁ U F₂)⁺. Hence, the decomposition of R (A, B, C, D) into R1 (A, C, D) and R2 (B, C) is not a dependency preserving decomposition.

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Dependency preserving decomposition - solved exercises 1

Dependency preserving decomposition - Dependency preserving decomposition solved exercises - How to verify that a decomposition is dependency preserving? - Steps to find dependency preserving decomposition - Dependency preserving decomposition examples

Dependency preserving decomposition

Consider a relation R (A, B, C, D) with the following set of functional dependencies;

F = {A → B, B → C, and C → D}.

Is the decomposition of R (A, B, C, D) into R1 (A, B, C) and R2 (C, D) a dependency preserving decomposition?

Solution:

The above said decomposition of R into R1 and R2 is a dependency preserving decomposition if (F₁ U F₂)⁺ = F⁺, where F₁ is set of FDs hold by R1, F₂ is set of FDs hold by R2, and F is the set of FDs hold by R.

Step 1: for R1, the derivable non-trivial functional dependencies are, A → B, and B → C. Hence, F₁ = {A → B, B → C}

Step 2: for R2, the derivable non-trivial functional dependency is, C → D. Hence, F₂ = {C → D}

(F₁ U F₂) = ({A → B, B → C} U {C → D}) = {A → B, B → C, C → D} = F.

F₁ U F₂ and F both have same set of functional dependencies. Hence, the decomposition of R (A, B, C, D) into R1 (A, B, C) and R2 (C, D) a dependency preserving decomposition.

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Dependency preserving decomposition - solved exercises

Dependency preserving decomposition - Dependency preserving decomposition solved exercises - How to verify that a decomposition is dependency preserving? - Steps to find dependency preserving decomposition - Dependency preserving decomposition examples

List of solved exercises/examples for Dependency preserving decomposition:

• Dependency preserving decomposition –Solved Exercise/Example 1

• Dependency preserving decomposition –Solved Exercise/Example 2

• Dependency preserving decomposition –Solved Exercise/Example 3

• Dependency preserving decomposition – Solved Exercise/Example 4

• Dependency preserving decomposition – Solved Exercise/Example 5

• Dependency preserving decomposition – Solved Exercise/Example 6

• Dependency preserving decomposition – Solved Exercise/Example 7

• Dependency preserving decomposition – Solved Exercise/Example 8

• Dependency preserving decomposition – Solved Exercise/Example 9

• Dependency preserving decomposition – Solved Exercise/Example 10