Given the relational schema \(R = (U, V, W, X, Y, Z)\) and the set of functional dependencies:
\[\{U \to V, U \to W, WX \to Y, WX \to Z, V \to X\}\]
Which of the following functional dependencies can be derived from the above set?
Answer
(A), (B), (D)
Explanation
Compute attribute closures.
- \(\{V, W\}^{+}\): \(V \to X\) adds \(X\), and then \(WX \to Y\) and \(WX \to Z\) add \(Y\) and \(Z\). So \(\{V, W\}^{+} = \{V, W, X, Y, Z\}\).
- \(\{W, X\}^{+} = \{W, X, Y, Z\}\), straight from the two given dependencies.
Hence \(VW \to YZ\) (A) and \(VW \to Y\) (D) hold, and \(WX \to YZ\) (B) is the union rule applied to \(WX \to Y\) and \(WX \to Z\). But \(U\) is not in \(\{V, W\}^{+}\): no dependency has \(U\) on its right-hand side, so only sets containing \(U\) determine it, and C cannot be derived.
Answer: A, B, D