Question

Consider a relation \(R(A, B, C, D, E)\) with the following three functional dependencies: \[ AB \rightarrow C, BC \rightarrow D, C \rightarrow E \] The number of superkeys in the relation is

Hint:

A superkey can be formed by adding any combination of attributes to a candidate key. The number of superkeys is determined by the number of attributes that can be added to the candidate key.

Answer 1

Correct Answer: 8

In the given relation, we need to calculate the number of superkeys. A superkey is a set of attributes that can uniquely identify a tuple in a relation.
We first determine the candidate keys:
- From \(C \rightarrow E\), we see that \(C\) determines \(E\).
- From \(AB \rightarrow C\), we see that \(AB\) determines \(C\), and hence \(AB\) determines both \(C\) and \(E\).
- From \(BC \rightarrow D\), we see that \(B\) and \(C\) together determine \(D\).

Thus, \(AB\) is a candidate key because it determines all other attributes (\(C, D, E\)).

Now, any superset of a candidate key is a superkey. Since \(AB\) is the only candidate key, the number of superkeys is given by the number of ways we can add additional attributes to \(AB\). We can add any combination of \(A, B, C, D, E\) to \(AB\), resulting in \(2^5 = 32\) superkeys.
Thus, the number of superkeys in the relation is: \[ \boxed{8} \]