Functional dependency: find candidate key

Question

In relation R = (ABCDEFGH) the set of functional dependencies that hold is
{BE -> GH, G -> FA, D -> C, F -> B}

What is a candidate key of R?

I found the answer can be BED, DEG, or FED. However, this question is a multiple choice question where I can't select multiple answers.

What is the method used and how do I use it here?

Answer 1

The set of candidate keys is {{BDE}, {DEF}, {DEG}}

A relation can have multiple candidate keys. However you can choose only one of them as a primary key.

Answer 2

To to find the candidate keys:

Form three columns: left, right and middle.

• In the left column, add the attributes which appear only on the left hand side of a FD or do not appear in any FD.
• In the right column, add the attributes which appear only on right hand side of FD.
• In the middle column, add attributes which appear both on right and left hand side of FD.

Explanation:

• Attributes in the left column must be included in every candidate key.
• Attributes in the right column must not be included in any candidate key.
• Attributes in the middle may or may not be included in candidate keys.

In this example, we get E, D in the left column, B, F, G in the middle and A, C, H in the right column.

Then, apply the Closure property:
BED+ = ABCDEFGH
since
BE->GH Given.
BE->FA Decomposition rule and transitivity rule and BE->G and G->FA.
BED->C Given since D->H.

Similarly DEF and DEG can be proved to be candidate keys.

Thus, all three are candidate keys.