GATE 2026 DA Q.59: solution and answer

GATE 2026 · DAQ.59 · 2 marks · NAT

Let there be two relations $X$ and $Y$ as shown. $X$ has three columns $P$ , $Q$ and $R$ . $Y$ has two columns $P$ and $S$ .

Relation X (columns P, Q, R): (P1, Q1, R1), (P2, Q2, R2), (P3, Q3, R2).

Relation Y (columns P, S): (P1, 10), (P1, 15), (P2, 20), (P3, 1).

Consider that the following tuple relational calculus expression is evaluated.

{t ∣ t \in X \land \exists z \in X (t [P] = z [P]) \land \exists m \in Y (m [P] = t [P] \land m [S] > 1)}

The number of tuples that will be returned is ______. (Answer in integer)

GATE 2026 DA question 59: Two tables side by side. — Figure from the GATE 2026 DA paper. It is part of the question.

Answer and solution

ANSWER

The number of tuples returned is 2.

Official answer: 2 to 3Source: Official answer key, GATE 2026 DA (IIT Guwahati)

The expression is ${t ∣ t \in X \land \exists z \in X (t [P] = z [P]) \land \exists m \in Y (m [P] = t [P] \land m [S] > 1)}$ . Every returned tuple $t$ comes from $X$ , and it must satisfy both existential conditions.
The condition $\exists z \in X (t [P] = z [P])$ is always true. Take $z = t$ , which is in $X$ because $t \in X$ . So it removes no tuple.
The real filter is $\exists m \in Y (m [P] = t [P] \land m [S] > 1)$ . $Y$ must contain a row with the same $P$ value as $t$ and with $S > 1$ .
For $t = (P 1, Q 1, R 1)$ : $Y$ has $(P 1, 10)$ and $(P 1, 15)$ . Both have $S > 1$ , so $t$ is returned.
For $t = (P 2, Q 2, R 2)$ : $Y$ has $(P 2, 20)$ and $20 > 1$ , so $t$ is returned.
For $t = (P 3, Q 3, R 2)$ : the only row of $Y$ with $P = P 3$ is $(P 3, 1)$ , and $1 > 1$ is false. So $t$ is not returned.
Tuple relational calculus returns a set of tuples $t$ taken from $X$ . The two matching rows of $Y$ for $P 1$ do not duplicate $(P 1, Q 1, R 1)$ , because it is listed once as a single tuple $t$ (unlike a join).
Exactly 2 tuples are returned: $(P 1, Q 1, R 1)$ and $(P 2, Q 2, R 2)$ .

KEY IDEA

The first existential is trivially true because $t$ can serve as $z$ . The second is a semi-join-like test on $X$ against $Y$ with the filter $S > 1$ . The result is a set of tuples from $X$ , so each qualifying tuple of $X$ appears once.

WATCH OUT FOR

Counting 3 tuples by treating $(P 3, 1)$ as a match even though $1 > 1$ is false. Counting 3 by treating the expression as a join, so that $P 1$ appears twice. A join would repeat P1, but this expression returns tuples $t$ of $X$ only.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 5 Relational algebra questions, with notes

GATE 2026 DA Q.59: solution and answer

Answer and solution

More Relational algebra questions