GATE CS 2012 Q13 — Propositional and First Order Logic

To translate the statement “Some real numbers are rational” into first-order logic, we analyze its components:

1. "Some" indicates the existential quantifier ($\exists x$), meaning there exists at least one element $x$ in the domain.
2. The statement asserts that this element $x$ satisfies two properties simultaneously:
   • $x$ is a real number: $\text{real}(x)$
   • $x$ is a rational number: $\text{rational}(x)$
3. To express that both conditions must hold true for the same $x$, we use the logical conjunction ($\wedge$).

Thus, the correct translation is:

Analysis of other options:

• Option A: $\exists x (\text{real}(x) \lor \text{rational}(x))$ states that there exists some $x$ which is either real or rational. This does not guarantee that any real number is rational.
• Option B: $\forall x (\text{real}(x) \to \text{rational}(x))$ translates to "All real numbers are rational".
• Option D: $\exists x (\text{rational}(x) \to \text{real}(x))$ is logically equivalent to $\exists x (\neg\text{rational}(x) \lor \text{real}(x))$. This is vacuously true if there exists anything that is not rational, which does not represent the original statement.

Therefore, the correct option is C.