I was just wondering what the truth value is for:
$\exists y \forall x (x^2 < y + 1)$. The domain of discourse is: $R$ X $R$.
The reason I believe this is false is x = y = 0. Which makes the condition $(x^2 < y + 1)$ false. Am I right?
$\forall y \exists x (x^2 < y + 1)$. The domain of discourse is: $R$ X $R$.
The reason I believe the second one is false once again is x = y = 0. Which makes the condition $(x^2 < y + 1)$ false. Am I right?
I'm sorry I'm new to this and I just am not sure of my answers... thank you for your help
For the first, the problem with your reasoning is this: you're basically saying that there is a $y$ for which it is not true. That doesn't mean that there couldn't exist a $y$ for which it IS true -- and that's what you're asked to consider!
If you want to show that the result "$\exists y\forall x(x^2<y+1)$" is false, you need to show that for EVERY $y$, the result "$\forall x(x^2<y+1)$" is false. Does that make sense?
For the second: again your result is correct, but your reasoning is no good! If you want to show that the result "$\forall y\exists x(x^2<y+1)$" is false, you need to show that there exists a $y$ such that "$\exists x(x^2<y+1)$" is false.