CS457 Syllabus & Progress

Equivalence of 2 sets of functional dependencies

Equivalence of 2 sets of functional dependencies

Definition: equivalence of sets of functional dependencies

Two sets of functional dependencies ℉₁ and ℉₂ are equal if:

Example of equivalent functional dependency set

Consider the following two sets of functional dependencies:

Relation R = (A, B, C, D, E, F) ℉₁ = { A → C AC → D E → AD E → F } ℉₂ = { A → CD E → ADF }

Question:

Are ℉₁ and ℉₂ equivalent ?

Does ℉₁ cover ℉₂ ???

Do the check for each functional dependency in ℉₂:

A → CD ∈ ℉₂ :

Compute A⁺ with respect to ℉₁: ℉₁ = { A → C AC → D E → AD E → F }
A⁺ = A (Initialization) = AC (A → C) = ACD (AC → D) Done Check: CD ⊆ ACD ?? Yes. Continue (next functional dependency)

E → ADF ∈ ℉₂ :

Compute E⁺ with respect to ℉₁: ℉₁ = { A → C AC → D E → AD E → F }
E⁺ = E (Initialization) = ADE (E → AD) = ADEF (E → F) = ACDEF (A → C) Done Check: ADF ⊆ ACDEF ?? Yes. Continue (no more functional dependency - DONE)

Conclusion:

℉₁ covers ℉₂

Does ℉₂ cover ℉₁ ???

Do the check for each functional dependency in ℉₁:

A → C ∈ ℉₁ :

Compute A⁺ with respect to ℉₂: ℉₂ = { A → CD E → ADF }
A⁺ = A (Initialization) = ACD (A → CD) Done Check: C ⊆ ACD ?? Yes. Continue (next functional dependency)

AC → D ∈ ℉₁ :

Compute AC⁺ with respect to ℉₂: ℉₂ = { A → CD E → ADF }
AC⁺ = AC (Initialization) = ACD (A → CD) Done Check: D ⊆ ACD ?? Yes. Continue (next functional dependency)

E → AD ∈ ℉₁ :

Compute E⁺ with respect to ℉₂: ℉₂ = { A → CD E → ADF }
E⁺ = E (Initialization) = EADF (E → ADF) = EADFC (A → CD) Done Check: AD ⊆ EADFC ?? Yes. Continue (next functional dependency)

E → F ∈ ℉₁ :

Compute E⁺ with respect to ℉₂: ℉₂ = { A → CD E → ADF }
E⁺ = E (Initialization) = EADF (E → ADF) = EADFC (A → CD) Done Check: F ⊆ EADFC ?? Yes. Continue (no next functional dependency - DONE)

Conclusion:

℉₂ covers ℉₁

So these two sets of FDs are equivalent ....