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?
Answer
(C), (D)
Explanation
A relation is in 3NF if, for every non-trivial FD \(X \to Y\), either \(X\) is a superkey or every attribute of \(Y\) is prime (part of some candidate key).
- R1: \(A\) and \(D\) appear on no right-hand side, so every key contains both, and \(\{A, D\}^{+} = \{A, B, C, D, E\}\). The only key is \(AD\). In \(D \to E\), \(D\) is not a superkey and \(E\) is not prime, so R1 is NOT in 3NF (it is not even in 2NF).
- R2: \(C\) appears on no right-hand side and \(C^{+} = \{A, B, C, D\}\), so \(C\) is the only key. In \(A \to D\), \(A\) is not a superkey and \(D\) is not prime: the transitive dependency \(C \to A \to D\). So R2 is NOT in 3NF.
Answer: C, D