Set of FDs are equivalent or not Exercise 11

How to prove that two sets of functional dependencies are equal or not - Proving equality of two sets of functional dependencies - How to find whether two sets of functional dependencies are equal or not? - Equivalent sets of functional dependencies

Exercise:

Let F = {A → C, AC → D, E → AD}, and let G = {A → CD, E → AHE}. Are they equivalent?

Solution:

To check whether two sets of functional dependency are equivalent, we have to verify every functional dependency (FD) of F is in G⁺ and every FD of G is in F⁺.

Theorem: If F ⊆ G⁺ and G ⊆ F⁺, then F and G are equivalent.

To check for F ⊆ G⁺

Let us take the first FD of F, A → C and find the closure using FDs from G as follows;

• A⁺ = ACD through the FD A → CD of G. The result includes the RHS attribute of FD A → C. Hence, A → C is in G⁺.

Let us do the same for other FDs of F;

• AC⁺ = ACD through the FD A → CD of G. The result includes the RHS attribute of FD AC → D. Hence, AC → D is in G⁺.

• E⁺ = AHECD through the FDs E → AHE and A → CD of G. The result includes the RHS attribute of FD E → AD. Hence, E → AD is in G⁺.

As all the FDs of F can be derived using G, F ⊆ G⁺ is true.

To check for G ⊆ F⁺

• A⁺ = ACD through the FDs A → C and AC → D of F. The result includes the RHS attribute of FD A → CD. Hence, A → CD is in F⁺.

• E⁺ = ACDE through the FDs E → AD, A → C and AC → D of F. The result does not include the RHS attribute of FD E → AHE. Hence, E → AHE is in F⁺.

As all the FDs of G cannot be derived using F, G ⊆ F⁺ is false.

Result:

We found that F ⊆ G⁺ is true but not G ⊆ F⁺. Hence, the given set of functional dependencies F and G are not equivalent (F ≠ G).  

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