# Determining truth value of quantified statements

Ok, sorry! I know I asked a question not but 1 hour ago, but I have one final question remaining about determining the truth value of a statement. I would again like confirmation of my answer for a base to go by for the rest of my questions.

Take this is as example 1:

U(x,y) means "2x + 3y = xy", where x and y are integers. Determine the truth value of the statement: ∀x∃y¬U(x,y)

My answer: The truth value is False because for example..if x=1 the final answer would not equal each other.

Example 2:

T(x,y) means "3x + 2y = xy", where x and y are integers. Determine the truth value of the statement: ∃y∀xT(x,y)

My answer: this is also false for the same reasons as example 1

Not going to lie, but all my answers are false so far...and that's a bit concerning...

I would love a guidance and / be pointed in the right direction!

• 1) ∀x∃y¬U(x,y) it is true, for example for x=0, y=1. 2) ∃y∀xT(x,y) it is true too. – Leox Oct 2 '13 at 22:51

The first sentence is true. For example, if $x\ne 0$ we can take $y=0$, and for $x=0$ we can take $y=1$.
The second sentence is indeed false. For if $x=0$ we would need $y=0$. And if $x=1$ we would need $y=-3$. So there is no single $y$ that works for all $x$.