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Which of the following is TRUE?

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

edited

@ Bikram sir can you plz explain meaning of option B

Arjun sir  Bikram  sir

B) "A relation R is in 3NF if every non-prime attribute of R is fully functionally dependent on every key of R"

means X->non_prime and X is key hence X is superkey also so it is in BCNF so also 3NF

plz check

Gate Ranker18 but here ans is c.according to u b is also correct

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

every non-prime attribute of R is fully functionally dependent on every key of R

it  gurantee there should not be the case of prime--->non-prime  other than Key--->non-prime

it does not say anything about  non-prime-->non prime, so it can occure here which for sure does not gurantee 3NF.

yes correct :)
edited by

for 3nf

LHS must be superkey OR

RHS must be prime attribute .According to second point  key------>non prime attribute

so here second condition(non prime attribute) of 3 nf fails.lets check first condition,key must be superkey hence 3nf,but nothing about non prime which violates 3 nf

Please explain difference between "fully functionally dependent" and "functionally dependent" Normal Form Set

(C) Every relation in $\text{BCNF}$ is also in $\text{3NF}.$ Straight from definition of $\text{BCNF}.$

by

Isn't B correct if we ignore the multivalued attributes?
Why is B incorrect , it also seems correct to me.

Nopes, that does not guarantee 3NF. For 3NF there should not be any FD X->Y such that Y-X is a non-key and X is not a super key.

Yes, got it .  :)

It should not have transitive dependency.
can anyone explain what option b says!

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

Lets take R(ABCD)  with AB is key

AB->C and C->D  here AB fully determine D i.e. AB-> CD

According to statement it is in 3NF but it is not. Because C->D form non prime -> non prime.

plz elaborate option B
B option says nothing about transitive dependency which should not be there in 3nf therefore B option not correct
In BCNF sometimes FD does not preserve. But In 3NF FD always prserve. So is it correct to say Every BCNF is 3NF?

For option B)

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

It only tells that Key=>Non-key but it does not guarantee that there will not be any transitive dependency i.e Non-key=>Non-Key FD is not present.

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

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 $C)$

edited
Ans is (C) Every relation in BCNF is also in 3NF. Straight from definition of BCNF.

@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 each key of the relation.

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 some key of the relation.

option C

here, in option B, it satisfies the requirement of 2NF. To be in 3NF no transitive dependency should ne there. Relation can have NP->NP FD satisfying statement B, but that will violate 3NF condition.

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