GATE DA 2025 | Question: 47

Question

​​Consider a database relation $\text{R}$ with attributes $\text{ABCDEFG}$, and having the following functional dependencies:

\[
\mathrm{A} \rightarrow \mathrm{BCEF} \quad \mathrm{E} \rightarrow \mathrm{DG} \quad \mathrm{BC} \rightarrow \mathrm{~A}
\]
Which of the following statements is/are correct?

• A. $\mathrm{A}$  is the only candidate key of $\mathrm{R}$ 
• B. $\mathrm{A}, \mathrm{BC}$ are the candidate keys of $\mathrm{R}$ 
• C. $\mathrm{A}, \mathrm{BC}, \mathrm{E}$ are the candidate keys of $\mathrm{R}$ 
• D. Relation $\mathrm{R}$ is not in Boyce-Codd Normal Form $\text{(BCNF)}$

Comments

Option B,D

A,BC arethe candidate keys

since BC-> A, BC will also be the candidate key

In option A is wrong because A is not the only one C.K

E->DG is a violation of BCNF so, not a BCNF

---

The key point to be noted in this question is that for a relation to be in BCNF, we must ensure that for every non-trivial functional dependency,  \( X \rightarrow Y \),  \( X \) must be a superkey.

In this case, \( A \) and \( BC \) are superkeys, but \( E \) is not a superkey.

\[
\therefore \text{The relation } R \text{ is not in BCNF.}
\]
 

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but we used to take minimal key ?
that would be A right?

although A and  BC both are  candidate key
but minimal would be one ... so option (A) and (D) should be answer ??
correct me if i am missing something..
 

Answer 1

\( R( A, B, C, D, E, F, G ) \)\[A \rightarrow BCEF\]\[E \rightarrow DG\]\[BC \rightarrow A\]

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\[BC \rightarrow A\]

\( A \rightarrow BCEF \),

\[BC^+ = \{ A, B, C, E, F \}\]

\( E \rightarrow DG \),\[BC^+ = \{ A, B, C, E, F, D, G \}\]

Since \( BC^+ \) contains all attributes of \( R \), \( BC \) is a candidate key.

\[A \rightarrow BCEF\]\( E \rightarrow DG \),\[A^+ = \{ A, B, C, E, F, D, G \}\]

Since \( A^+ \) contains all attributes of \( R \), \( A \) is a candidate key.

---

 

A relation BCNF if for every functional dependency \( X \rightarrow Y \), \( X \) is a superkey.

 

\[A \rightarrow BCEF\]\[E \rightarrow DG\]\[BC \rightarrow A\]

 

\( E \rightarrow DG \) is a dependency where \( E \) is not a candidate key or a superkey (since \( E^+ = \{ E, D, G \} \neq R \)). Since this violates BCNF, the relation is \textbf{not in BCNF}.
 

Answer 2

R(A, B, C, D, E, F)

Functional Dependencies : A -> BCNF; E -> DG; BC -> A

 

Step 1 :- Find Candidate Keys

(A)⁺ = {A, B, C, D, E ,F, G} // A is Candidate Key 

(BC)⁺ = {B, C, A, E, F, G, D} // BC is Candidate Key

(E)⁺ = {E, D, G} // E is not a Candidate Key

∴ Candidate Key = {A, BC} // Satisfied Option B

 

Step 2 :- Check whether it is in BCNF form or not

A -> BCNF => A is a Super Key (BCNF satisfied)

E -> DG => E is neither a Super Key nor it follows Trivial Dependency (Y ⊆ X).

∵ E -> DG violated BCNF rules,

∴ It is not in BCNF // Satisfied Option D

 

Correct Answer :

B) A, BC are the candidate keys of R. 

D) Relation R is not in Boyce-Codd Normal Form (BCNF)