GATE CS 2026 Set 1 Q.43: Relational Algebra and Tuple Relational Calculus

Consider a relational database schema with two relations $R(P, Q)$ and $S(X, Y)$ .

Let $E = \{\langle u \rangle \mid \exists v \, \exists w \, \langle u, v \rangle \in R \wedge \langle v, w \rangle \in S\}$ be a tuple relational calculus expression.

Which one of the following relational algebraic expressions is equivalent to $E$ ?

$\Pi_P(R \bowtie_{R.P = S.X} S)$

$\Pi_P(S \bowtie_{S.X = R.Q} R)$

$\Pi_P(R \bowtie_{R.P = S.Y} S)$

$\Pi_P(S \bowtie_{S.Y = R.Q} R)$

Official answer and solution

Official answer: B
From: Official GATE 2026 answer key

Solution

Match the positions of variables $u, v, w$ in $\langle u, v \rangle \in R(P, Q)$ and $\langle v, w \rangle \in S(X, Y)$ with their respective attribute names.

In the tuple relational calculus expression $E = \{\langle u \rangle \mid \exists v \, \exists w \, \langle u, v \rangle \in R \wedge \langle v, w \rangle \in S\}$ :

$\langle u, v \rangle \in R(P, Q)$ binds $u = R.P$ and $v = R.Q$ .
$\langle v, w \rangle \in S(X, Y)$ binds $v = S.X$ and $w = S.Y$ .
The shared variable $v$ enforces the join condition $S.X = R.Q$ , and the output tuple $\langle u \rangle$ projects attribute $P$ ( $R.P$ ).

Thus $E$ is equivalent to $\Pi_P(S \bowtie_{S.X = R.Q} R)$ .

Final answer

Option (B)

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: 43 of 65
Section: Computer Science and Information Technology

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