Consider the following relational schema along with all the functional dependencies that hold on them. $R1(A, B, C, D, E): \{ D \to E, EA \to B, EB \to C \}$ $R2(A, B, C, D): \{ A \to D, A \to B, C \to A \}$ Which of the following statement(s) is/are TRUE?

To check whether a relation is in Third Normal Form (3NF), every non-trivial functional dependency $X \to Y$ must satisfy at least one of the following conditions:

1. $X$ is a superkey, or

2. $Y$ is a prime attribute (part of a candidate key).

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### Analysis of $R1(A, B, C, D, E)$:

• Given FDs:$\{ D \to E, EA \to B, EB \to C \}$

• Attributes $A$ and $D$ do not appear on the right-hand side of any functional dependency, so they must be part of any candidate key.

• Computing the attribute closure of $AD$:

  $$(AD)^+ = \{A, D, E, B, C\}$$

• Therefore, the only candidate key is $AD$.

• Prime attributes: $\{A, D\}$

• Non-prime attributes: $\{B, C, E\}$

• For the functional dependency $D \to E$:

  • $D$ is not a superkey ($(D)^+ = \{D, E\} \neq \{A, B, C, D, E\}$).

  • $E$ is not a prime attribute.

• Thus, $R1$ violates the condition for 3NF, meaning $R1$ is NOT in 3NF.

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### Analysis of $R2(A, B, C, D)$:

• Given FDs:$\{ A \to D, A \to B, C \to A \}$

• Computing attribute closures:

  • $(C)^+ = \{C, A, D, B\} = \{A, B, C, D\} \implies C$ is a candidate key.

  • $(A)^+ = \{A, D, B\} \neq \{A, B, C, D\}$

• The only candidate key is $C$.

• Prime attributes: $\{C\}$

• Non-prime attributes: $\{A, B, D\}$

• For the functional dependency $A \to D$ (or $A \to B$):

  • $A$ is not a superkey.

  • $D$ (and $B$) is not a prime attribute.

• Thus, $R2$ violates the condition for 3NF, meaning $R2$ is NOT in 3NF.

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Therefore, statements (C) and (D) are TRUE.