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For database relation $R(A,B,C,D)$ where the domains of $A,B,C$ and $D$ include only atomic values, only the following functional dependencies and those that can be inferred from them are:

$A \rightarrow C$

$B \rightarrow D$

The relation $R$ is in

1. First normal form but not in second normal form
2. Both in first normal form as well as in second normal form
3. Second normal form but not in third normal form.
4. Both in second normal form as well as in third normal form.

### 1 comment

Original Source of this question: GATE 1997

The key for relation will be composite {A,B}

A+ = {A,C}

B+ = {B,D}

{A,B}+ = {A,B,C,D} so it is the key

prime attributes = {A,B} (members of key}

non prime attributes = {C,D}

since all attributes contain atomic values it is in first normal form

non prime attribute C is partially dependent on  key (A --> C}

non prime attribute D is partially dependent on  key (B --> D}

for a relation to be in second normal form

all non prime attributes should be fully dependent on key

but here non prime attributes are partially dependent. Clearly it is not in second normal form

First normal form but not in second normal form.

Option 1.

obviously it is in 1NF.

Now,Here primary key is AB

2NF doesnot allow partial dependency. It occurs when a non-key attribute of a table in a database is dependent on the value of only a part of the table’s primary key but not the entire primary key.

Here C depends on A,, that is partial.. same for B -> D

so not in 2NF

here A -> C and B->D no singleton attribute derives all other Attribute.But if we take AC->BD

so closure of ac={AC}+={A,B.C,D}

so AB is the only candidate key of the relation!

now as 2nf says,a ttributes, should be fully dependent on the Key

But here C,and D are partiallly dependent

So it is in 1NF

Candidate key is AB and

According to 2NF Non Prime Attributes should be Fully Functional Dependent on any key

So NPA = C,D

whereas A,B is not key so Not 2NF just 1NF option A

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