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Let $R_1 \left(\underline{A}, B, C\right)$ and $R_2\left(\underline{D}, E \right)$ be two relation schema, where the primary keys are shown underlined, and let C be a foreign key in $R_1$ referring to $R_2$. Suppose there is no violation of the above referential integrity constraint in the corresponding relation instances $r_1$ and $r_2$. Which of the following relational algebra expressions would necessarily produce an empty relation?

1. $\Pi_D (r_2) - \Pi_C (r_1)$

2. $\Pi_C (r_1) - \Pi_D (r_2)$

3. $\Pi_D \left(r_1 \bowtie_{C \neq D}r_2\right)$

4. $\Pi_C \left(r_1 \bowtie_{C = D}r_2\right)$

What if the field 'C' of relation R1 has NULLs  for some of the records (I believe FKs can have NULLs). Will option B always return empty set?

ans (B)

C in R1 is a foreign key referring to the primary key D in R2. So, every element of C must come from some D element.
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Why not A?
because we can add values in D , that wont violate the referential integrity constraints..

so there can be a case when there are extra elements in D that are not  there in C therefore option A will  be wrong.