P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.
-$$\forall y \exists x P ( x , y )$$

Question

Quizplus · Accepted Answer

The statement claims that for every integer $y$, there exists an integer $x$ such that $x + 2y = xy$. However, if we choose $y = 0$, the equation simplifies to $x = 0$, which means $x$ must be 0 to satisfy the equation for $y = 0$. But if $y$ is any non-zero value, the equation does not guarantee a solution for $x$ in integers without further conditions. The statement is false because it does not hold for every possible integer value of $y$, as demonstrated by counterexamples where no integer $x$ satisfies the equation for certain values of $y$.

P(x, Y) Means "X + 2y = Xy," Where X ∀y∃xP(x,y)\forall y \exists x P ( x , y )∀y∃xP(x,y)

P(x, Y) Means "X + 2y = Xy," Where X $\forall y \exists x P ( x , y )$