What is the correct translation of the following statement into mathematical logic? "Some real numbers are rational" A. (forall x)(real(x) -> rational(x)) B. (forall x)(real(x) ^ rational(x)) C. (exists x)(real(x) ^ rational(x)) D. (exists x)(real(x) -> rational(x))

GATE 2012 · Discrete Mathematics · First Order Logic · easy

Answer: C. (exists x)(real(x) ^ rational(x))

1. Identify quantifier and connective: 'Some real numbers are rational' -> there exists an x such that x is real AND x is rational. Formula: (exists x)(real(x) ^ rational(x)).
2. Eliminate wrong options: A uses forall with ->, which means 'all reals are rational' — false (pi is real but irrational). B uses forall with ^, which claims everything in the domain is both real and rational — false. D uses exists with ->, which is logically different and would be vacuously true for non-reals. Only C, (exists x)(real(x) ^ rational(x)), correctly captures 'some real is rational'.