GATE CS 2000 Q2.26 — Tuple Calculus and SQL

To determine which pair is not equivalent, we must evaluate the expressions under the given condition: all comparisons with a null value are treated as false.

Let us analyze the behavior of each option when $x$ is null:

Option A: $x = 5$ and $\text{not}(\text{not}(x = 5))$

• If $x$ is null:
  • $x = 5 \implies \text{false}$
  • $\text{not}(\text{not}(x = 5)) \implies \text{not}(\text{not}(\text{false})) \implies \text{not}(\text{true}) \implies \text{false}$
  Both expressions evaluate to false.
• If $x$ is not null: Both expressions naturally yield the same boolean value.

Thus, this pair is equivalent.

Option B: $x = 5$ and $x > 4 \text{ and } x < 6$ (where $x$ is an integer)

• If $x$ is null:
  • $x = 5 \implies \text{false}$
  • $(x > 4 \text{ and } x < 6) \implies \text{false and false} \implies \text{false}$
  Both expressions evaluate to false.
• If $x$ is an integer and not null: The only integer satisfying $x > 4$ and $x < 6$ is $5$. Thus, both expressions evaluate to true when $x = 5$, and false otherwise.

Thus, this pair is equivalent.

Option C: $x \neq 5$ and $\text{not}(x = 5)$

• If $x$ is null:
  • $x \neq 5$: Since this is a comparison with a null value, it evaluates to false.
  • $\text{not}(x = 5)$: The inner comparison $x = 5$ evaluates to false. Thus, $\text{not}(\text{false})$ evaluates to true.
  Since one expression evaluates to false and the other to true, they are not equivalent.

Thus, this pair is not equivalent.