Question 31

Relation $R(A, B, C, D, E)$ has functional dependencies:

1. Attributes $A$ and $B$ do not appear on the right-hand side of any FD. Therefore, $A$ and $B$ must be part of every candidate key.
2. Compute the attribute closure of $\{A, B\}$:
   $$(AB)^+ = \{A, B\} \cup \{C\} = \{A, B, C\}$$
   Using $BC \to D$: $(AB)^+ = \{A, B, C, D\}$
   Using $C \to E$: $(AB)^+ = \{A, B, C, D, E\}$

Since $(AB)^+ = \{A, B, C, D, E\}$ and neither $A$ nor $B$ alone can derive the other attributes, $AB$ is the unique candidate key of $R$.
3. A superkey is any superset of a candidate key. Since $AB$ is the only candidate key, any superkey must contain both $A$ and $B$.
4. The remaining 3 attributes $\{C, D, E\}$ can be included or excluded independently in $2^3 = 8$ ways.

Thus, there are exactly 8 superkeys.