Translating this predicate logic statement?

Question

Given that $P(x, y): “x + 2y = xy”$, where $x$ and $y$ are integers. How do we translate

$\lnot$$\forall$$x$ $\exists$$y$ $\lnot$$P(x,y)$

and what's the truth value of this?

Answer 1

$¬∀ ∃ ¬(,)$

Use the negated quantifiers,

$∃ ∀ (,)$.

This statement is True when there exists x, for all y, P(x,y) holds.

Suppose that $x=x_0$ exists,

Require to Prove: for all y, $P(x_0,y)$ holds for all $y \in Z$.

Given that (,):“+2=”, where and are integers.

Require to Prove: $(_0,):“_0+2=_0”$ holds for all $y \in Z$.

Require to Prove: $(_0,):“_0+2=_0”$ holds for all $y \in Z$.

Require to Prove: $(_0,):“\frac{_0}{x_0-2}=, (x_0≠2)”$ holds for all $y \in Z$.

This is not true as there exists y = 1, $(_0,)$ doesn't hold as $x_0≠x_0 -2$.

Hence, the statement is False.