B) "Some are not my friends and are perfect"

C) "Some are not my friends and are not perfect"

D) "There exist no one who is my friend and is perfect"

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55 votes

Best answer

2

None of my friends are perfect can be written an "all of my friends are not perfect".

Now all of my friends are not perfect could be written as:

$\forall x (F(x) \implies \lnot P(x))$

$\forall x (\lnot F(x) \cup \lnot P(x))$

By de Morgan's law: -

$\lnot \exists(F(x) \cap P(x))$

Which is equivalent to option D.

Sir, what is the right way to interpret it?

4

41 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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