selft doubt

Question

can we consider  , proper subset of candidate key union proper subset of another candidate key

as a proper subset of candidate key.

Answer 1

Consider the relation scheme R(A, B, C, D, E, H) and the set of functional dependencies-

A → B

BC → D

E → C

D → A

• Determine all essential attributes of the given relation.
• Essential attributes of the relation are- E and H.
• So, attributes E and H will definitely be a part of every candidate key.

So candidate keys are AEH, BEH, and DEH. 
 

Now, we can see A is proper subset of AEH. 

And B is proper subset of BEH (another candidate key). 
 

A union B is AB, which is not a proper subset of any candidate key. So, we can’t consider the given statement to be true. 

Answer 2

This is Some time Yes , Sometime No ,So Totally(in General) No

Relation R(A, B, C)

No FDs (other than trivial).

 Find candidate keys

• Since there are no FDs, you need n−1 attributes 

So candidate keys = AB, BC, AC

 Take proper subsets

• Proper subset of AB = {A}

• Proper subset of BC = {C}

Union = AC

Is AC a candidate key?

 YES, 

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Relation R(A, B, C, D)

FDs:

• A → B

• C → D

• D → B, candidate keys = AC and AD

Take proper subsets

• Proper subset of AC = {A}

• Proper subset of AD = {D}

Now union = {A,D}

Is AD a candidate key?

NO, because:

• From A → B

• From D → B

• But we CANNOT derive C

• So AD⁺ = {A,B,D} ≠ all attributes

So union is NOT a candidate key.