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52 votes
52 votes

What is the logical translation of the following statement?

"None of my friends are perfect."

  1. $∃x(F (x)∧ ¬P(x))$
  2. $∃ x(¬ F (x)∧ P(x))$
  3. $ ∃x(¬F (x)∧¬P(x))$
  4. $ ¬∃ x(F (x)∧ P(x))$
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8 Answers

Best answer
65 votes
65 votes
  1. some of my friends are not perfect
  2. some of those who are not my friends are perfect
  3. some of those who are not my friends are not perfect
  4. NOT (some of my friends are perfect) / none of my friends are perfect

Correct Answer: $D$

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48 votes
48 votes
  • $F\left(x\right): x \text{ is my friend.}$
  • $P\left(x\right):\text{x is perfect.}$

$\text{“None of my friends are perfect"}$ can be written like

$\forall x[F(x)\implies\neg P(x)]$
$\equiv \forall x[\neg F(x)\vee \neg P(x)]$
$\equiv \forall x\neg[F(x)\wedge p(x)]$
$\equiv \neg\exists x[F(x)\wedge P(x)]$

So, the answer is D.

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9 votes
9 votes
"None of my friends are perfect."  

It is NOT the complement of "All of my friends are perfect"   So A is not the answer. (A frequently done mistake)

It is the complement of "At least one of my friend is perfect"  So D is the answer.
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