Relational Theory – Assessing Third Normal Form Statement

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Consider the following statements:

A relation R, is in 3NF if every non-prime attribute of R, is fully functionally dependent on some key of R.

State True or False.

The answer given is true. But I think its false.

Example:
Let R(A,B,C) be a relation with functional dependencies {A→B, B→C}.

Here A is the candidate key and both B and C are fully functionally dependent on A since A→B and also A→C right?

But this relation is clearly not in 3NF. Where am I wrong?

As per the definition I found here, the set of attributes X will be fully functionally dependent on the set of attributes Y if the following conditions are satisfied:

X is functionally dependent on Y and
X is not functionally dependent on any subset of Y.

Here C is functionally dependent on A and also C is not functionally dependent on any subset of A. So this would imply that C is fully functionally dependent on A right?

Best Answer

The exercises says:

A relation R is in 3NF, if every non-prime attribute of R, is fully functionally dependent on some key of R.

As you have noted, that definition is imprecise. Let’s try to correct it:

A relation R is in 3NF, if every non-prime attribute of R is fully functionally dependent on some (candidate) key and on nothing else.

Of course such definition should also exclude trivial dependencies or derived dependencies in which the left part contains superflous attributes.

Related Solutions

Why is this relation in 3NF

The mistake is in your understanding of transitive dependency. From wikipedia: Transitive dependency

In mathematics, a transitive dependency is a functional dependency which holds by virtue of transitivity. A transitive dependency can occur only in a relation that has three or more attributes. Let A, B, and C designate three distinct attributes (or distinct collections of attributes) in the relation. Suppose all three of the following conditions hold:

1. A → B
2. It is not the case that B → A
3. B → C

Then the functional dependency A → C (which follows from 1 and 3 by the axiom of transitivity) is a transitive dependency.

In your case though, (2) does not hold:

1. U → T                                  -- correct
2. It is not the case that T → U          -- wrong
3. T → V                                  -- correct

Therefore {V} is NOT transitively dependent on {U}.

Questions regarding second normal form

If all your dependencies are those you have shown: {A->GH, B->IJ, C->A, F->B, FC->DK, K->E} then you have come to the wrong conclusion.

Your only candidate key is CF. The C and F are not candidate keys on their own.

Therefore, the F->B dependency (and the C->A as well) means that the relation violates 2NF.

For the other question, if you had for example these dependencies:

BC -> AF
AF -> BC
 F -> DE

where the candidate keys are BC and AF, then again the F -> DE would mean that the relation violates 2NF. To be in 2NF would mean that there is no dependency on any part of any candidate key.

Best Answer

Related Solutions

Why is this relation in 3NF

Questions regarding second normal form

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