What would be the best normalised relational model for the Entity E?

Entity: E (A, B, C, D)

Functional Dependencies :

A, B → C, D

C → B

Question

What would be the best normalised relational model for the Entity E?

Entity:  E (A, B, C, D)

Functional Dependencies :

A, B → C, D

C →  B

BlackTom AI · Accepted Answer

The question asks for the best normalised relational model for the Entity E with attributes A, B, C, D and the given functional dependencies: A,B → C,D and C → B.

First, restating the problem in my own words: we want a decomposition of the relation E(A,B,C,D) into smaller relations that achieve proper normalization, ideally BCNF or at least 3NF, while preserving dependencies and ensuring a lossless join.

Option 1: E1(A, C, D) and E2(C, B).
- Here we decompose into two relations: R1 with attributes {A, C, D} and R2 with attributes {C, B}, joined on C. This matches the standard BCNF decomposition step for the FD C → B: separate B into its own relation with C as the key, leaving A, C, D in another relation.
- In R2, the only FD that concerns these attributes is C → B, and C is a key for R2, so this is BCNF for that piece. In R1, the attributes are A, C, D, and what FDs apply? From AB → C, D we would need B, which is not in R1, so this projection does not introduce a violation there; effectively A and C determine D within the scope given by the original dependencies, and the decomposition leaves A, C, D in a structure where the remaining dependency AB → C, D is satisfied via the join with R2. Overall, this decomposition is commonly recognized as a lossless-join, dependency-preserving BCNF decomposition for the given set of FDs.
- Therefore, this option correctly captures the essential normalization step: isolating the C → B dependency in its own relation and keeping A, C, D together where they remain functionally related through the remaining FDs.

Option 2: E1 (A, C, D) and E2 (C, B).
- This option is essentially identical to Option 1, just repeated in the list. The same analysis applies: a BCNF/3NF-leaning decomposition that preserves dependencies and allows a lossless join by joining on C. Since the problem statement presents the same schema split in two places, this option is also a correct representation of the best normalised model according to the given dependencies.

Option 3: E(A, B, C, D).
- Keeping the original four-attribute relation in one piece does not reflect normalization. It must be checked against the given FDs: C → B violates BCNF because C is not a superkey in the full relation (C+ = {C,B} does not include A or D). This indicates that the original schema is not BCNF and should be decomposed. Thus, this option is not the best normalised model.
- In addition, it does not address lossless-join concerns or dependency preservation, since BCNF requires decomposition that handles the problematic FD C → B. So this option is not appropriate as the best normalised form.

Option 4: E(A, C, B, D).
- This option appears to reorder attributes but, functionally, the same issue as Option 3 persists: the presence of C → B means that a single relation with all four attributes cannot be in BCNF. The order of attributes does not affect the normalization status; the FD C → B still causes a violation in BCNF. Consequently, this is not the best normalised model.

In summary, the reasoning shows that the decomposition into two relations on the attributes {A, C, D} and {C, B} is the appropriate BCNF/3NF-friendly form given the dependencies, since it isolates the determinant C and its dependent B, while keeping A and D with C in a way that respects the remaining dependency AB → C, D through the join. The two identical first options provide the correct normalised model, whereas the latter two options fail to achieve the BCNF decomposition and do not reflect the necessary normalization step for the given dependencies.

What would be the best normalised relational model for the Entity E? Entity: E (A, B, C, D) Functional Dependencies : A, B → C, D C → B

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