GATE CS 2000 Q2.25 — Tuple Calculus and SQL

To understand when the SQL query is guaranteed to produce the same result as relation $r$, let's analyze the operations performed by the query:

SELECT DISTINCT w, x 
FROM r, s

1. Cartesian Product (FROM r, s): This pairs every tuple in $r(w, x)$ with every tuple in $s(y, z)$.
   • If $s$ is empty (contains $0$ tuples), the Cartesian product $r imes s$ will be empty, resulting in an empty output. Thus, for the output to match $r$, $s$ must be non-empty.
   • If $s$ is non-empty (contains at least one tuple), then every tuple $(w_i, x_i) \in r$ will appear at least once in the Cartesian product.
2. Projection and Duplicate Elimination (SELECT DISTINCT w, x): This projects the attributes $w, x$ and removes any duplicate rows from the final output.
   • If the original relation $r$ already contained duplicate tuples, the DISTINCT keyword would eliminate them, making the query result different from $r$.
   • Therefore, to guarantee the result is identical to $r$, $r$ must have no duplicate tuples.

Combining these two conditions, the query is guaranteed to return the same result as $r$ if and only if $r$ has no duplicates and $s$ is non-empty.

Thus, Option A is the correct choice.