GATE CS 2013 Q.27 (CS Q.27): Propositional and First-Order Logic

What is the logical translation of the following statement?

"None of my friends are perfect."

$\exists x (F(x) \land \neg P(x))$

$\exists x (\neg F(x) \land P(x))$

$\exists x (\neg F(x) \land \neg P(x))$

$\neg \exists x (F(x) \land P(x))$

Official answer and solution

Solution

Translate the natural language statement into formal predicate logic by analyzing the meaning of the universal denial.

Let the universe of discourse be people. Define predicates:

The statement "None of my friends are perfect" asserts that there is no person who is both my friend and perfect. Formally:

\neg \exists x (F(x) \land P(x))

Equivalently, this statement can be written universally as:

\forall x (F(x) \rightarrow \neg P(x))

Among the given options, option (D) is $\neg \exists x (F(x) \land P(x))$ , which directly translates the original sentence.

Final answer

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