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Which one is the correct translation of the following statement into mathematical logic?

“None of my friends are perfect.”

  1. $\neg\:\exists\:x(p(x)\land q(x))$
  2. $\exists\:x(\neg\:p(x)\land q(x))$
  3. $\exists\:x(\neg\:p(x)\land\neg\:q(x))$
  4. $\exists\:x(p(x)\land\neg\:q(x))$
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let write a logic for "All my friends are perfect"

$\vee x(q(x)\rightarrow p(x))$

take negation of above "none of my friends are perfect"

$\sim(\vee x(q(x)\rightarrow p(x)))$

$\sim\vee x (\sim q(x)\vee p(x))))$

$Ǝx(q(x)\wedge \sim p(x))$

 

Correct me if I am wrong.
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“None of my friends are perfect.”

It says that It is a false that i have friend and he is perfect 

 $\sim \exists x(P(x) \wedge q(x)))$

Answer:

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