Logic Basic

Question

I have a proposition here

Not Everyone is perfect

The above statement also mean there is someone who is perfect

So can i write

∃xP(x) where P(x) is x is perfect

And in book it is written

∼∀x P(x)

Which one is correct and where am i wrong ?

Comments

I think not everyone is perfect means No one is perfect.That's why answer is ∼∀x P(x) 
More clarification is required for me also.. :(

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I think Not everyone can be ambigious . @Arjun  sir guide here

Answer 1

The above statement also mean there is someone who is perfect

NO. The only thing that you can infer from the above statement is that "There exists someone whose is not perfect."

It may be the case that everybody is not perfect.

This is one of the reasons why plain English is not used for mathematical relations, which tends to lead to ambiguity.

$\neg \forall (x) P(x) \implies \exists (x) \neg (P(x))$

Comments

Thanks mojo - jojo :P I agree totally with your solution :)

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Welcome :)