GATE CS 1999 Q1.19 — Tuple Calculus and SQL

The given tuple relational calculus (TRC) expression is:

$$\{t \mid t \in r \land (t[A] = 10 \land t[B]=20)\}$$

This expression selects all tuples $t$ from relation $r$ where both conditions $t[A] = 10$ and $t[B] = 20$ are satisfied simultaneously. In relational algebra, this is represented by the selection operation:

$$\sigma_{(A=10 \land B=20)}(r)$$

Let us analyze the given options to find the equivalent expression:

• Option A: $\sigma_{(A=10 \lor B=20)} (r)$ uses the logical OR ($\lor$) operator, which is incorrect.
• Option B: $\sigma_{(A=10)} (r) \cup \sigma_{(B=20)} (r)$ is equivalent to $\sigma_{(A=10 \lor B=20)} (r)$ because the union operator ($\cup$) corresponds to the logical OR ($\lor$).
• Option C: $\sigma_{(A=10)} (r) \cap \sigma_{(B=20)} (r)$ selects tuples that satisfy $A=10$ and satisfy $B=20$. The intersection operator ($\cap$) corresponds to the logical AND ($\land$). Thus, it is equivalent to $\sigma_{(A=10 \land B=20)}(r)$.
• Option D: $\sigma_{(A=10)} (r) - \sigma_{(B=20)} (r)$ selects tuples where $A=10$ but $B \neq 20$, which is incorrect.

Therefore, the correct equivalent expression is Option C.