Question

What is the logical translation of the following statements? "None of my friends are perfect"

• A. $\exists x\left(F\right(x)\wedge \neg P(x\left)\right)$
• B. $\exists x\left(\neg F\right(x)\wedge P(x\left)\right)$
• C. $\exists x\left(\neg F\right(x)\wedge \neg P(x\left)\right)$
• D. $\neg \exists x\left(F\right(x)\wedge P(x\left)\right)$

Answer 1

The correct option is D $\neg \exists x\left(F\right(x)\wedge P(x\left)\right)$

None of my friends are perfect i.e., all of my friends are not perfect
$\forall x\left(\right(F\left(x\right)\to \neg P\left(x\right))$
$\forall x\left(\neg F\right(x)\vee \neg P(x\left)\right)$
$\neg \exists x\left(F\right(x)\wedge P(x\left)\right)$
Alternatively
$\exists x\left(F\right(x)\wedge P(x\left)\right)$ gives
there exist some of my friends who are perfect.
$\neg \exists x\left(F\right(x)\wedge P(x\left)\right)$
there does not exits any friend who is perfect i.e none of my friends are perfect.
So (d) is correct option.