Consider the schema R = (A, B, C, D, E, F) on which the following functional dependencies hold:

A ➝ B

B, C ➝ D

E ➝ C

D ➝ A

What are the candidate keys of R?

Concept:

Candidate key: it is the set of attributes which uniquely identifies a relation. It is also known as superkey with no repeated attributes.

Explanation:

Given relation R = {A, B, C, D, E}

A ➝ B

B, C ➝ D

E ➝ C

D ➝ A

Find the closure of keys given. If all the attributes of relation are present in the closure, then it will be the candidate key of that relation.

Option1: AEF, BEF and DEF

(AEF)⁺ = {A, E, F, B, C, D}

(BEF)⁺ = {B, E, F, C, D, A}

(DEF)⁺ = {D, E, F, C, A, B}

These are the candidate keys of given relation.

Option2: AEF, BEF and BCF

(AEF)⁺ = {A, E, F, B, C, D}

(BEF)⁺ = {B, E, F, C, D, A}

(BCF)⁺ = {B, C, F, D, A}        / E is not present in the closure. Not the candidate key

Option3: AE and BE

(AE)⁺ = {A, E, C, B, D, }       /F is not present, not a candidate key

(BE)⁺ = {B, E, C, D, A}        /F is not present, not a candidate key

Option4: AE, BE and DE

(AE)⁺ = {A, E, C, B, D, }       /F is not present, not a candidate key

(BE)⁺ = {B, E, C, D, A}        /F is not present, not a candidate key

(DE)⁺ = {D, E, A, C, B}        /F is not present, not a candidate key