Question

Suppose the following functional dependencies hold on a relation \(U\) with attributes \(P, Q, R, S,\) and \(T\):
\[ P \rightarrow QR, RS \rightarrow T \] Which of the following functional dependencies can be inferred from the above functional dependencies?

• A. \(PS \rightarrow T\)
• B. \(R \rightarrow T\)
• C. \(P \rightarrow R\)
• D. \(PS \rightarrow Q\)

Hint:

Use Armstrong's axioms: decomposition, augmentation, and transitivity to infer new functional dependencies.

Answer 1

The Correct Option is A, C, D

Step 1: Use decomposition on \(P \rightarrow QR\).
From \(P \rightarrow QR\), by decomposition, we get: \[ P \rightarrow Q \text{and} P \rightarrow R. \] Hence, statement (C) is true.

Step 2: Infer \(PS \rightarrow T\).
From \(P \rightarrow R\), by augmentation with \(S\), we obtain: \[ PS \rightarrow RS. \] Given \(RS \rightarrow T\), by transitivity: \[ PS \rightarrow T. \] Hence, statement (A) is true.

Step 3: Infer \(PS \rightarrow Q\).
From \(P \rightarrow Q\), by augmentation with \(S\), we get: \[ PS \rightarrow Q. \] Thus, statement (D) is true.

Step 4: Eliminate incorrect option.
There is no dependency that allows inferring \(R \rightarrow T\) without \(S\). Hence, (B) is false.