@ Bikram sir can you plz explain meaning of option B

Which of the following is **TRUE**?

- Every relation in $\text{3NF}$ is also in $\text{BCNF}$
- A relation $\text{R}$ is in $\text{3NF}$ if every non-prime attribute of $\text{R}$ is fully functionally dependent on every key of $R$
- Every relation in $\text{BCNF}$ is also in $\text{3NF}$
- No relation can be in both $\text{BCNF}$ and $\text{3NF}$

### 10 Comments

@ Gate Ranker18 in case B) * non-prime-->non prime* dependency is possible which is not the case in 3NF(transitive dependency)

rajoramanoj acc to me both B and C are correct

reena_kandari it is **X-> non_prime** not **non-prime-->non prime**

## 4 Answers

### 12 Comments

$\text{1. Every relation in 3NF is also in BCNF}$

For a $3NF$ relation to be in $BCNF$, that has to has satisfied this condition-

$\text{for each dependency }$ $X \rightarrow Y$, $\text{X should be the super key.}$

So, this statement is wrong

$\text{2. A relation R is in 3NF if every non-prime attribute of R is fully functionally dependent}$ $\text{on every key of R}$

This is a straightforward definition of $2NF$. If a relation is in $1NF$ and in that relation each & every non-prime attributes has to be fully functionally dependent on each key of the relation.

& for a $2NF$ relation to be in $3NF$, that has to satisfy the condition which states, no non-prime attributes should be transitively dependent on candidate key or we can say that, no non-prime attribute should be determined by anything but a super key

Let's take an example:

$R(ABCDE)$ and $F = \{BC \rightarrow ADE, A \rightarrow BCDE, D \rightarrow E\}$

The candidate keys will be = $\{ A, BC\}$

Prime Attribute = $A,B,C$

Non-Prime Attribute = $D,E$

Here, all the non-prime attributes$(D,E)$ are fully functionally dependent on each key of $R$

But here, $E$(non-prime attribute) can be determined by $D$ and $D$ is not the candidate key. So, $E$ is transitively dependent on the candidate keys of $R$ and moreover, $D$ is not the super key also.

$\therefore$ This relation is not in $3NF$. For making it to $3NF$ we have to break the relation into $2$ parts: $R_1(ABCD), F=\{ \{A \rightarrow BCD\}, BC \rightarrow AD\}$ & $R_2(DE), F = \{D \rightarrow E\}$

This option is also wrong.

$\text{3. Every relation in BCNF is also in 3NF}$

Obviously, For a relation to be in $BCNF$, it has to be in $3NF$ first.

$\color{Green}{\therefore \text{This statement is Correct}}$

$\text{4. No relation can be in both BCNF and 3NF}$

Yes, there can be some relation which is in $3NF$ and as well as $BCNF$

and if a relation is in $BCNF$ then it is also in $3NF$

So, this statement is also wrong.

Correct option is $C)$

### 2 Comments

Answer about confusion option B is:- https://gateoverflow.in/33464/3nf-true-false

@Subarna Das Ma'am I think there is a correction in your statement 2.

This is a straightforward definition of 2NF. If a relation is in 1NF and in that relation each & every non-prime attributes has to be fully functionally dependent on

key of the relation.each

It should be

This is a straightforward definition of 2NF. If a relation is in 1NF and in that relation each & every non-prime attributes has to be fully functionally dependent on each

key of the relation.some

# Correct Answer is (C)

There is no confusion in option A and D that they are wrong

And there also should not any confusion in option C that is option C is correct.

In option (B) some students have confusion

So I explain that how option B is wrong

**According to the definition of 2NF and 3NF**

**2 NF** – A relation R is in second normal form (2NF) if and only if it is in 1NF and every non-key attribute is **fully dependent** on the primary key.

Note that it does not put any restriction on** the non-prime to non-prime attribute dependency**.

For Example

**A--->B, B--->C , A--->C here A is key and relation is in 2NF.**

Here A is Primary Key and every non key attributes i.e. B and C fully dependent on A

**But here B--->C is also a functional dependency which violate 3NF**

3 NF – A relation R is in third normal form (3NF) if and only if it is in 2NF and every **non-key attribute is non-transitively dependent** on the primary key.

**In the option it is given 2NF definition not of 3NF so It is wrong**