GATE CS 2019 Q55 — Relational Model: Relational Algebra

To find the number of tuples returned by the query, we evaluate the two parts of the set difference expression separately.

Part 1: $T_1 = \Pi_X (\sigma_{(P.Y=R.Y \wedge R.V='V2')} (P \times R))$

• We first filter relation $R$ for $V = \text{V2}$: $\sigma_{V='V2'}(R) = \{(\text{Y3}, \text{V2}), (\text{Y2}, \text{V2})\}$.
• Next, we join this result with $P$ on $Y$. The only matching tuple in $P$ with $Y \in \{\text{Y2}, \text{Y3}\}$ is $(\text{X2}, \text{Y2}, \text{Z2})$.
• Projecting on $X$ gives: $T_1 = \{\text{X2}\}$.

Part 2: $T_2 = \Pi_X (\sigma_{(Q.Y = R.Y \wedge Q.T > 2)} (Q \times R))$

• We first filter relation $Q$ for $T > 2$: $\sigma_{T>2}(Q) = \{(\text{X1}, \text{Y2}, 5), (\text{X1}, \text{Y1}, 6)\}$.
• We join these tuples with $R$ on $Y$. Since both $Y = \text{Y2}$ and $Y = \text{Y1}$ exist in $R$, both tuples join successfully.
• Projecting on $X$ gives: $T_2 = \{\text{X1}\}$.

Set Difference:

$$T_1 - T_2 = \{\text{X2}\} - \{\text{X1}\} = \{\text{X2}\}$$

The final relation contains exactly 1 tuple.