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"

Dark Mode

56 votes

Best answer

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.

0