What is the negation of the statement below x Px Hey I got t

What is the negation of the statement below?
x ~P(x)

Hey I got this question wrong but i\'m trying to figure out where my logic went wrong. The statement \"x ~P(x)\" should translate to \"All of x is NOT in P(x)\", correct? So wouldn\'t the negation be \"All of x IS in P(x)\" which is x P(x)?

The correct answer is listed as x P(x)

Thanks!

Solution

The approach you are using is universal quantifier.

When we negate a quantifier statement, we negate all the quantities first, from left to right (keeping the same order), then we negative the statement.

Ax ~P(x) = 3x P(x)

To negate a statement: we flip to , and then negate the predicate inside. That is,
• the negation of x : P(x) is x : P(x).
This, incidentally, is where the term “counterexample” comes from.

If x : P(x) is false,
then x : P(x) — and the x that exists to satisfy P(x) is the counterexample to the claim
x : P(x).
On the other hand, to negate x : P(x), we must claim that P(x) fails to hold for any possible x. So again we flip the quantifier and then negate the predicate:
• the negation of x : P(x) is x : P(x).