What is the correct translation of the following statement into mathematical logic? “Some real numbers are rational” A) ∃ x (real(x) ∨ rational(x)) B) ∀ x (real(x) → rational(x)) C) ∃ x (real(x) ∧ rational(x)) D) ∃ x (rational(x) → real(x))

Official answer: C) ∃ x (real(x) ∧ rational(x)). Explanation: The statement "Some real numbers are rational" asserts that there exists at least one entity x such that x is both a real number and a rational number. In predicate logic, this existential statement is translated as: ∃ x (real(x) ∧ rational(x)) - Option A states that there exists an x which is either real or rational. - Option B states that all real numbers are rational. - Option D states that there exists an x such that if it is rational, then it is real (which is trivially true for anything that is not rational). Thus, C is the correct option.

A) ∃ x (real(x) ∨ rational(x))

B) ∀ x (real(x) → rational(x))

D) ∃ x (rational(x) → real(x))

Propositional and First-Order Logic

GATE CSE 2012 · Q13

MCQ1 markEasy

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What is the correct translation of the following statement into mathematical logic? “Some real numbers are rational”

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Official answer: C) ∃ x (real(x) ∧ rational(x))

Propositional and First-Order Logic

The statement "Some real numbers are rational" asserts that there exists at least one entity $x$ such that $x$ is both a real number and a rational number.

In predicate logic, this existential statement is translated as:

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

Option A states that there exists an $x$ which is either real or rational.
Option B states that all real numbers are rational.
Option D states that there exists an $x$ such that if it is rational, then it is real (which is trivially true for anything that is not rational).

Thus, C is the correct option.

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This Propositional and First-Order Logic previous year question is Q13 of the GATE CSE 2012 paper. Find more GATE CSE PYQs on Propositional and First-Order Logic in Engineering Mathematics, or browse all GATE CSE PYQs.

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Propositional and First-Order Logic in GATE CSE

Propositional and First-Order Logic has 18 GATE CSE PYQs across 16 papers from 2011 to 2026. 6 of them are 1-mark MCQs like this one, and it is the only one in GATE CSE 2012.

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