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Which one of the following statements is $\text{FALSE}$?

1. Any relation with two attributes is in $\text{BCNF}$
2. A relation in which every key has only one attribute is in $\text{2NF}$
3. A prime attribute can be transitively dependent on a key in a $\text{3 NF}$ relation
4. A prime attribute can be transitively dependent on a key in a $\text{BCNF}$ relation

Any relation with two attributes is in $\text{BCNF} \Rightarrow$ This is true. It is trivial

A relation in which every key has only one attribute is in $\text{2NF}\Rightarrow$ This is true. As it is not possible to have Partial Functional Dependency !

A prime attribute can be transitively dependent on a key in a $\text{3NF}$ relation $\Rightarrow$ This is true. As For $\text{3NF}$ to be violated we need something like Key $\Rightarrow$ Non Key, Non Key $\Rightarrow$ Non key. $\text{3NF}$ definition says that for functional dependency $x\rightarrow y,$ either $x$ should be key or $y$ should be prime attribute. Then we can have something like Key $\Rightarrow$ Non Key, Non key $\Rightarrow$ Prime Attribute, resulting in Transitive $\text{FD}$ on Prime Attribute, still in $\text{3NF}$.

$\text{LHS}$ must be always key, so No Transitive dependency is allowed.

This example is not in 3nf , since D and C both are non prime attribute. 3nf does not support transtive dependency between non prime attribute
It's impossible that a non prime attribute derives a prime attribute. Then that non prime attribute is prime attribute
How can a non prime determine prime?

If it so then it must be prime

(d)

Defn from wiki:

The 3NF version of the definition is weaker than Date's BCNF variation, as the former is concerned only with ensuring that non-key attributes are dependent on keys. Prime attributes (which are keys or parts of keys) must not be functionally dependent at all; they each represent a fact about the key in the sense of providing part or all of the key itself. (It should be noted here that this rule applies only to functionally dependent attributes, as applying it to all attributes would implicitly prohibit composite candidate keys, since each part of any such key would violate the "whole key" clause.)

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thnx @Digvijay Panday all doubt clear from your comment :)
Whenever we have non-prime->prime dependency then that non-prime on LHS becomes prime too, because as RHS is prime attribute it's definitely part of some key now if we remove that part from key and added the LHS of above FD we will get new key and hence it will also become prime, so there will never be non-prime derives prime dependency, friends if find any mistake please correct

A) possible. If a relation has 2 attribute, then there are 1:1 dependency,So, it will be in BCNF

B) If every key has one attribute , it must have no partial dependency. So, it will be in 2 NF

C) prime attribute can be transitively dependent on a key - i.e. key dependency will be non prime -> prime . But no non prime to non prime dependency will be there. So, it could be in 3NF

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can u play explain  option c ... what u said I did not get
By defination of 3NF, X->Y, X can Super key or Y is prime attribute, hence prime attribute can be transitevely dependent on a Key.
correct it to prime -> non-prime in your option-C explanation.

option d is false 