Both statements are False. Thinking them True is a Misconception.
Counterexample for Statement 1:
\[
R(A, B, C, D, E)
\]
Functional dependencies:
\[
AB \;\to\; CDE
\]
\[
CD \;\to\; E
\]
\[
D \;\to\; E
\]
- Candidate key = \( AB \).
- Prime attributes = \( A, B \).
- Non-prime attributes = \( C, D, E \).
- Here, \( CD \to E \) is a partial dependency (since \( D \to E \)), but it does not violate 2NF, because 2NF only forbids when some CK partially determines 'non-prime attribute'. Here 'Non-prime attributes' are paritally determining 'non-prime attribute' so it is allowed in 2NF.
- Thus, the relation is in 2NF, so Statement 1 is false.
Counterexample for Statement 2:
\[
R(A, B, C, D)
\]
Functional dependencies:
\[
AB \;\to\; CD
\]
\[
CD \;\to\; AB
\]
\[
A \;\to\; C
\]
- Candidate keys = \( AB, CD \).
- Prime attributes = \( A, B, C, D \).
- Since all attributes are prime, there are no 3NF violations.
- Notice that \( AB \to A \) and \( A \to C \) create a transitive dependency \( AB \to C \), but it does not violate 3NF, because the dependent attribute \( C \) is prime. 3NF violates when some CK transitively determines 'non-prime attribute' but here it is determining 'prime attribute' so allowed.
- Thus, the relation is in 3NF, so Statement 2 is false.
So, answer is D.
Misconceptions in Partial Dependency, Transitive Dependency: https://youtu.be/xqIm-Y3oVDk
