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

To solve this problem, we need to determine which functional dependencies can be logically inferred from the given set of dependencies. The given functional dependencies for relation \(U\) with attributes \(P, Q, R, S,\) and \(T\) are:

• \(P \rightarrow QR\)
• \(RS \rightarrow T\)

We need to infer possible dependencies from these and check them against the provided options.

Let's analyze each option:

1. \(PS \rightarrow T\)

We know that \(RS \rightarrow T\). If \(P\) is known to functionally determine \(R\), then combined with \(S\), we can say \(PS \rightarrow T\). Since \(P \rightarrow R\) is a part of \(P \rightarrow QR\), by transitivity (if \(X \rightarrow Y\) and \(Y \rightarrow Z\), then \(X \rightarrow Z\)), \(PS \rightarrow T\) can be logically inferred.

2. \(R \rightarrow T\)

Given \(RS \rightarrow T\), for \(R \rightarrow T\) to be true, \(S\) should not add any information to \(T\), which cannot be determined from the given dependencies. Hence, \(R \rightarrow T\) cannot be inferred.

3. \(P \rightarrow R\)

The functional dependency \(P \rightarrow QR\) implies that \(P\) determines both \(Q\) and \(R\) together. Therefore, \(P \rightarrow R\) can be inferred directly as \(R\) is a subset of \(QR\).

4. \(PS \rightarrow Q\)

Since it is given that \(P \rightarrow QR\), we know \(P\) alone can determine \(Q\). Hence, adding any additional information such as \(S\) will not affect that. By augmentation (if \(X \rightarrow Y\), then \(XZ \rightarrow YZ\)), we get \(PS \rightarrow Q\).

Thus, the correct functional dependencies that can be inferred are:

• \(PS \rightarrow T\)
• \(P \rightarrow R\)
• \(PS \rightarrow Q\)