GATE CS 2026 Set 1 Q33 — Tuple Calculus and SQL

To find the relational algebra expression equivalent to the given Tuple Relational Calculus (TRC) expression, let us analyze its components step-by-step.

1. Analyze the Relation Schemas

• Relation $R(P, Q)$ has attributes $P$ (first position) and $Q$ (second position).
• Relation $S(X, Y)$ has attributes $X$ (first position) and $Y$ (second position).

2. Deconstruct the TRC Expression

The given TRC expression is:

• First term: $\langle u, v\rangle \in R$
  Since $R$ has attributes $(P, Q)$, this implies: $$u = R.P \quad \text{and} \quad v = R.Q$$
• Second term: $\langle v, w\rangle \in S$
  Since $S$ has attributes $(X, Y)$, this implies: $$v = S.X \quad \text{and} \quad w = S.Y$$

3. Establish the Equivalence

Since the variable $v$ represents both $R.Q$ and $S.X$, we must equate them:

The output tuple contains only $\langle u\rangle$, which corresponds to the attribute $P$ of relation $R$. Therefore, we project only $P$ after performing a theta-join between $R$ and $S$ on the condition $S.X = R.Q$:

This matches Option B.