Question:
Given a relation \(R(A, B, C, D, E)\) with functional dependencies \(F = \{A \rightarrow B,\ BC \rightarrow D,\ E \rightarrow C,\ D \rightarrow A\}\), which of the following is a Candidate Key for \(R\)?
कार्यात्मक निर्भरताओं \(F = \{A \rightarrow B,\ BC \rightarrow D,\ E \rightarrow C,\ D \rightarrow A\}\) के साथ एक संबंध \(R(A, B, C, D, E)\) दिया गया है। निम्नलिखित में से कौन सी \(R\) के लिए एक उम्मीदवार कुंजी (Candidate Key) है?
Options:
- \(AE\)
- \(AB\)
- \(CD\)
- \(BE\)
- \(AE\)
- \(AB\)
- \(CD\)
- \(BE\)
Correct Answer: a)
Explanation:
Notice that \(E\) does not appear on the right-hand side of any dependency in \(F\), so \(E\) must be part of every candidate key.
Let’s test \((AE)^+\):
\((AE)^+ = \{A, E\} \rightarrow \{A, B, E\} \text{ (using } A \rightarrow B) \rightarrow \{A, B, C, E\} \text{ (using } E \rightarrow C) \rightarrow \{A, B, C, D, E\} \text{ (using } BC \rightarrow D)\).
Since \((AE)^+\) contains all attributes and no proper subset of \(AE\) does, \(AE\) is a valid candidate key.
Let’s test \((AE)^+\):
\((AE)^+ = \{A, E\} \rightarrow \{A, B, E\} \text{ (using } A \rightarrow B) \rightarrow \{A, B, C, E\} \text{ (using } E \rightarrow C) \rightarrow \{A, B, C, D, E\} \text{ (using } BC \rightarrow D)\).
Since \((AE)^+\) contains all attributes and no proper subset of \(AE\) does, \(AE\) is a valid candidate key.
चूंकि \(E\) किसी भी निर्भरता के दाएं पक्ष में नहीं आता, अतः \(E\) प्रत्येक उम्मीदवार कुंजी में होना अनिवार्य है। \((AE)^+ = \{A, B, C, D, E\}\) सभी विशेषताओं को निर्धारित करता है, अतः \(AE\) उम्मीदवार कुंजी है।