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GATE CS 2026 Set 1 · Databases

Question 65

NAT2 marksNo negative markingEasy (estimate)

Question

Consider a relational database schema with a relation R(A,B,C,D)R(A, B, C, D). If {A,B}\{A, B\} and {A,C}\{A, C\} are the only two candidate keys of the relation RR, then the number of superkeys of relation RR is ______. (answer in integer)

Official answer and solution
Official answer
6
From
Official GATE 2026 answer key

Solution

Use the principle of inclusion-exclusion to count all subsets of {A,B,C,D}\{A, B, C, D\} that contain at least one candidate key.

A subset S⊆{A,B,C,D}S \subseteq \{A, B, C, D\} is a superkey of RR if and only if SS contains {A,B}\{A, B\} or SS contains {A,C}\{A, C\}:

  • Number of subsets containing {A,B}\{A, B\}: 24−2=42^{4 - 2} = 4 ({A,B},{A,B,C},{A,B,D},{A,B,C,D}\{A,B\}, \{A,B,C\}, \{A,B,D\}, \{A,B,C,D\})
  • Number of subsets containing {A,C}\{A, C\}: 24−2=42^{4 - 2} = 4 ({A,C},{A,B,C},{A,C,D},{A,B,C,D}\{A,C\}, \{A,B,C\}, \{A,C,D\}, \{A,B,C,D\})
  • Number of subsets containing both {A,B}\{A, B\} and {A,C}\{A, C\} (i.e., containing {A,B,C}\{A, B, C\}): 24−3=22^{4 - 3} = 2 ({A,B,C},{A,B,C,D}\{A,B,C\}, \{A,B,C,D\})

By inclusion-exclusion, the total number of superkeys is 4+4−2=64 + 4 - 2 = 6.

Final answer

6

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper
GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question
65 of 65
Section
Computer Science and Information Technology

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