GATE CS 2012 Q.43 (CS Q.43): Relational Algebra and Tuple Relational Calculus

Suppose $R_1(A, B)$ and $R_2(C, D)$ are two relation schemas. Let $r_1$ and $r_2$ be the corresponding relation instances. $B$ is a foreign key that refers to $C$ in $R_2$ . If data in $r_1$ and $r_2$ satisfy referential integrity constraints, which of the following is ALWAYS TRUE?

$\Pi_B(r_1) - \Pi_C(r_2) = \emptyset$

$\Pi_C(r_2) - \Pi_B(r_1) = \emptyset$

$\Pi_B(r_1) = \Pi_C(r_2)$

$\Pi_B(r_1) - \Pi_C(r_2) \ne \emptyset$

Official answer and solution

Official answer: A
From: Official GATE 2012 answer key

Solution

Apply the relational definition of foreign keys and referential integrity constraints.

Let $B$ in relation $r_1$ be a foreign key referencing primary key $C$ in relation $r_2$ .
The referential integrity constraint requires that for every tuple in $r_1$ , the foreign key value in $B$ must either be NULL or match an existing value in attribute $C$ of $r_2$ .
In terms of relation instances (assuming non-null foreign key values as modeled in relational algebra):

\Pi_B(r_1) \subseteq \Pi_C(r_2)

In set algebra, set $X$ is a subset of set $Y$ ( $X \subseteq Y$ ) if and only if the difference $X - Y$ is empty:

\Pi_B(r_1) - \Pi_C(r_2) = \emptyset

Thus, statement (A) is ALWAYS TRUE.

Final answer

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

Source

Paper: GATE 2012 Computer Science and Information Technology (CS)
Question: 43 of 65, printed as CS Q.43
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source