An organization needs to maintain database having five attributes A, B, C, D,…

Given the set of functional dependencies F = {A → BC, D → E, BC → D, A → D}.

Definition: An FD X → Y is redundant if it is implied by the other FDs in the set. To test redundancy, remove the FD and compute the closure of X using the remaining dependencies.

1. Check A → D:

   Remove A → D and compute A+ from the remaining FDs:

   • Start: A+ = {A}

   • From A → BC add B and C: A+ = {A, B, C}

   • From BC → D add D: A+ = {A, B, C, D}

   • Since D ∈ A+, A → D is implied by the remaining dependencies, so A → D is redundant.

2. Check BC → D:

   Remove BC → D and compute BC+ from the remaining FDs:

   • Start: BC+ = {B, C}

   • None of the remaining FDs (A → BC, D → E, A → D) let you derive D from BC, so D ∉ BC+.

   • Thus BC → D is not implied by the others and is not redundant.

3. Check D → E:

   Remove D → E and compute D+ from the remaining FDs:

   • Start: D+ = {D}

   • No remaining FD allows deriving E from D, so E ∉ D+.

   • Therefore D → E is not redundant.

4. Check A → BC:

   Remove A → BC and compute A+ from the remaining FDs:

   • Start: A+ = {A}

   • From A → D add D (if A → D is retained), and from D → E add E, so A+ may include D and E but does not include B or C.

   • Since BC ∉ A+, A → BC is not implied by the others and is not redundant.

Final answer: A → D is the redundant functional dependency because it is implied by A → BC and BC → D.