GATE CS 2012 Q.13 (CS Q.13): Propositional and First-Order Logic

What is the correct translation of the following statement into mathematical logic?

"Some real numbers are rational"

$\exists x (\text{real}(x) \lor \text{rational}(x))$

$\forall x (\text{real}(x) \rightarrow \text{rational}(x))$

$\exists x (\text{real}(x) \land \text{rational}(x))$

$\exists x (\text{rational}(x) \rightarrow \text{real}(x))$

Official answer and solution

Official answer: C
From: Official GATE 2012 answer key

Solution

Translate the natural language existential assertion into first-order predicate logic using conjunction.

The sentence to translate is: "Some real numbers are rational".

"Some" denotes the existential quantifier $\exists$ .
The statement asserts that there exists at least one object $x$ that is both a real number and a rational number:

\exists x (\text{real}(x) \land \text{rational}(x))

Note that using an implication $\exists x (\text{real}(x) \rightarrow \text{rational}(x))$ is incorrect because it would be vacuously true if there existed any object that was not a real number (since $\text{False} \rightarrow P$ is true). Existential statements across categories are universally formulated with conjunction ( $\land$ ).

Therefore, option (C) is the correct translation.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2012 Computer Science and Information Technology (CS)
Question: 13 of 65, printed as CS Q.13
Section: Computer Science and Information Technology

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