GATE CS 2014 Set 2 Q.31 (CS Q.21): Functional Dependencies and Normalization

The maximum number of superkeys for the relation schema $R(E, F, G, H)$ with $E$ as the key is _____.

Your answer

Official answer and solution

Solution

Use the definition of superkey as any superset of a candidate key in the relation schema.

Definition of Superkey:
A superkey for relation schema $R$ is any set of attributes containing a candidate key as a subset.
Schema Attributes:
The relation schema is $R(E, F, G, H)$ with 4 attributes, and $E$ is given as the key.
Counting Superkeys Containing $E$ :
Any superkey must include attribute $E$ . The remaining 3 attributes $\{F, G, H\}$ may either be included or excluded.
$\text{Number of subsets of } \{F, G, H\} = 2^3 = 8.$
Listing All 8 Superkeys:
- Single-attribute: $\{E\}$
- Two-attribute: $\{E, F\}, \{E, G\}, \{E, H\}$
- Three-attribute: $\{E, F, G\}, \{E, F, H\}, \{E, G, H\}$
- Four-attribute: $\{E, F, G, H\}$

Total number of superkeys = $8$ .

Final answer

Common trap

Counting all possible subsets of all 4 attributes ( $2^4 = 16$ ) without requiring key attribute E to be present.

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