is this right truth value

Question

Let $P(x,y)$ be the predicate $2x+y = xy$, where the domain of discourse for $x$ and $y$ is integers. Determine the truth value of each statement.

$P(-1,1)$ : true

$\exists xP(x,y)$ : true

$\exists yP(4,y)$ : false

$\forall yP(2,y)$ : false

$\forall x \exists y(x,y)$ : false

$\exists y \forall x(x,y)$ : false

Have I answered correctly or have not?

Please guide me if I'm wrong. thank you.

Answer 1

Interpretation of third statement:

Expression $\exists y P(4,y)$ means: There is at least one $y$ such that $P(4,y)$ is satisfied. Check:

$2\cdot 4 +y=4y\\\\ -3y=-8\\\\ y=\dfrac{8}{3}$

So, third statement is false because only possible value for $y$ is a fraction.

Other statements are answered correct as well.

Answer 2

Indeed, you've answered correctly in all but the second statement.

$∃xP(x,y)$ corresponds to the statement "There exists an $x$ such that $2x+y = xy$. The truth-value is indeterminate, because we know nothing about $y$; it is unbound, whether such a y exists, etc. to make the statement true or false.

Suggestion: When you submit your answers, if space is available, show your work to justify your conclusions. If nothing else, record the statements, your conclusions, and the reasoning that led you to the conclusions; those notes will handy notes to have.