GATE CS 2012 Q43 — Relational Model: Relational Algebra

By definition, a referential integrity constraint ensures that every value of a foreign key in a referencing relation must exist in the primary key of the referenced relation.

Here, $B$ is a foreign key in $R_1$ that refers to the primary key $C$ in $R_2$. Therefore, every value of $B$ present in the instance $r_1$ must also be present in the column $C$ of the instance $r_2$.

In terms of relational algebra, this subset relationship is expressed as:

Using set theory, if $X \subseteq Y$, then the set difference $X - Y$ is empty ($X - Y = \varnothing$). Substituting our projections:

Hence, Option A is always true.

Analysis of other options:

• Option B: $\prod_{C}(r_{2}) - \prod_{B}(r_{1}) = \varnothing$ is only true if every primary key value in $r_2$ is referenced by some tuple in $r_1$. This is not mandatory.
• Option C: $\prod_{B}(r_{1}) = \prod_{C}(r_{2})$ is a special case of Option B and is not always true.
• Option D: $\prod_{B}(r_{1}) - \prod_{C}(r_{2}) \neq \varnothing$ represents a violation of referential integrity (where a foreign key value exists in $r_1$ but not in $r_2$).