Consider the following relational schema along with all the functional dependencies that hold on them.
R1(A,B,C,D,E):{D→E,EA→B,EB→C}
R2(A,B,C,D):{A→D,A→B,C→A}

Which of the following statement(s) is/are TRUE?

The correct option is: R₂ is NOT in 3NF

To determine whether the relational schemas are in Third Normal Form (3NF), we need to analyze their candidate keys and check the 3NF conditions for their functional dependencies.

1. Third Normal Form (3NF) Definition:
A relation schema $R$ is in 3NF if, for every non-trivial functional dependency $X\to Y$ holding on $R$, at least one of the following conditions is satisfied:
- $X$ is a superkey of $R$, or
- $Y$ is a prime attribute of $R$ (i.e., $Y$ is a member of some candidate key of $R$).

2. Analyzing R₂(A, B, C, D):
The given set of functional dependencies for R₂ is:
1. $A\to D$
2. $A\to B$
3. $C\to A$

Step A: Find the Candidate Keys of R₂
Let us examine the attributes that do not appear on the right-hand side of any functional dependency in R₂. The attribute $C$ only appears on the left-hand side of the functional dependency $C\to A$ and never on the right-hand side of any dependency. Therefore, $C$ must be a part of every candidate key of R₂.
Let us compute the attribute closure of $C$ (denoted as $C^{+}$):
$C^{+}=\left\{C\right\}$
Using the functional dependency $C\to A$, we add $A$ to the closure:
$C^{+}=\{C,A\}$
Using the functional dependencies $A\to D$ and $A\to B$, we add $D$ and $B$ to the closure:
$C^{+}=\{A,B,C,D\}$
Since the closure $C^{+}$ contains all attributes of R₂, $C$ is a candidate key. Because $C$ is essential and itself determines all attributes, $C$ is the unique candidate key of R₂.

Step B: Identify Prime and Non-Prime Attributes
- Prime attribute (part of the candidate key): {C}
- Non-prime attributes (not part of the candidate key): {A, B, D}

Step C: Verify the 3NF Conditions for R₂
Now we check the functional dependencies of R₂ against the 3NF rules:
- For $C\to A$:
The determinant $C$ is a candidate key (superkey), so this dependency satisfies the 3NF criteria.

- For $A\to D$:
1. The determinant $A$ is not a superkey.
2. The dependent $D$ is a non-prime attribute (since the only candidate key is $C$).
Therefore, $A\to D$ violates the rules of 3NF.

- For $A\to B$:
1. The determinant $A$ is not a superkey.
2. The dependent $B$ is a non-prime attribute.
Therefore, $A\to B$ also violates the rules of 3NF.

Since R₂ contains functional dependencies that do not satisfy either condition of the 3NF definition, the relational schema R₂ is NOT in 3NF.