Let R (ABCDEFGH) be a relation schema and F be the set of dependencies F = {A → B, ABCD → E, EF → G, EF → H and ACDF →EG}. The minimal cover of a set of functional dependencies isa)A → B, ACD → E, EF → G, EF → H and ACDF → Gb)A → B, ACD → E, EF → G, EF → H and ACDF → Ec)A → B, ABCD → E, EF → H and EF → Gd)A → B, ACD → E, EF → G, and EF → HCorrect answer is option 'D'. Can you explain this answer?

Question

Let R (ABCDEFGH) be a relation schema and F be the set of dependencies F = {A &rarr; B, ABCD &rarr; E, EF &rarr; G, EF &rarr; H and ACDF &rarr;EG}. The minimal cover of a set of functional dependencies isa)A &rarr; B, ACD &rarr; E, EF &rarr; G, EF &rarr; H and ACDF &rarr; Gb)A &rarr; B, ACD &rarr; E, EF &rarr; G, EF &rarr; H and ACDF &rarr; Ec)A &rarr; B, ABCD &rarr; E, EF &rarr; H and EF &rarr; Gd)A &rarr; B, ACD &rarr; E, EF &rarr; G, and EF &rarr; HCorrect answer is option 'D'. Can you explain this answer?

Answer

-> B, B -> C, C -> D, D -> E, E -> F, F -> G, G -> H}.

The closure of a set of attributes X under the set of functional dependencies F, denoted as X+, is the set of all attributes that are functionally determined by X based on the given functional dependencies.

To find the closure of a set of attributes under a set of functional dependencies, we can use Armstrong's axioms, which are a set of inference rules for functional dependencies. The axioms are:

1. Reflexivity: If Y is a subset of X, then X -> Y.
2. Augmentation: If X -> Y, then XZ -> YZ for any set of attributes Z.
3. Transitivity: If X -> Y and Y -> Z, then X -> Z.

Using these axioms, we can find the closure of a set of attributes by repeatedly applying the axioms until no more attributes can be added to the closure.

In this case, let's find the closure of attribute set A.

Step 1: Start with the attribute set A.
Closure(A) = A.

Step 2: Apply the functional dependencies to add attributes to the closure.

Using A -> B, we add B to the closure.
Closure(A) = AB.

Using B -> C, we add C to the closure.
Closure(A) = ABC.

Using C -> D, we add D to the closure.
Closure(A) = ABCD.

Using D -> E, we add E to the closure.
Closure(A) = ABCDE.

Using E -> F, we add F to the closure.
Closure(A) = ABCDEF.

Using F -> G, we add G to the closure.
Closure(A) = ABCDEFG.

Using G -> H, we add H to the closure.
Closure(A) = ABCDEFGH.

Step 3: No more attributes can be added to the closure.
Final closure of A under the given functional dependencies is ABCDEFGH.

Answer

Use the union rule to replace

EF → G and EF → H

EF → GH

F = { A → B ABCD → E EF → GH ACDF → EG }

B is extraneous in ABCD → E because B ∈ ABCD and {A → B, ABCD → E, EF → GH, ACDF → EG}

logically implies {A → B, ACD → E, EF → GH, ACDF → EG}.

This is because every ACD → E.

This FD can be derived using Armstrong’s Axioms from A → B and ABCD → E via transitivity rule

So remove B from ABCD → E.

F = { A → B ACD → E EF → GH ACDF → EG }

E is extraneous in ACDF → EG because E ∈ EG and {A → B, ACD → E, EF → GH, ACDF → G}

logically implies {A → B, ACD → E, EF → GH, ACDF → EG}

remove E from ACDF → EG

F = { A → B ACD → E EF → GH ACDF → G}

G is extraneous in ACDF → G. Note that ACDF → G is already implied by ACD → E and EF → GH in F

remove ACDF → G from F.

None of the remaining FD's in F have extraneous attributes so the minimal cover is

A → B, ACD → E, EF → G, EF → H.

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