Given a relational schema R(ABCDEFGH) in first normal form. For the set of…

Disclaimer : In the original exam paper, both Option A and Option D were technically correct (Option A was a redundant augmentation of Option D). To remove ambiguity and make this a single-correct question, Option A has been modified to an incorrect dependency.

Concept: A functional dependency X → Y is logically implied by F exactly when Y belongs to the attribute closure X+.

Armstrong’s transitivity rule derives X → Z from X → Y and Y → Z. Augmentation derives XZ → YZ from X → Y, and decomposition may then retain a subset of the right-hand side.

Application: Compute the relevant closures under F = {A → B, A → C, CG → H, B → H, G → F}.

1. Begin with A+ = {A}.

2. A → B adds B, and A → C adds C, giving A+ = {A, B, C}.

3. Because B is present, B → H adds H, so A+ = {A, B, C, H}. Therefore A → H is implied.

4. For AC+, A → B again adds B and B → H adds H, giving AC+ = {A, B, C, H}. Therefore AC → H is also implied. Equivalently, augment A → H with C to obtain AC → HC and decompose it to AC → H.

Cross-check / contrast: The other determinant closures do not reach H.

• C+ = {C}; it does not contain H.

• G+ = {G, F}; it does not contain H.

• A+ and AC+ both contain H.

Result: The displayed single-correct item is ambiguous because both A → H and AC → H are logically implied. The question or its option set needs correction before publication.