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Let $Q(x,y)$ be the statement $y = 2x +1$ what are the values of the following. The universe is $Z^+$ {1,2,3,....}.

(a) $\forall x\exists y Q(x,y)$ This is true

(b) $\exists x \forall y Q(x,y)$ This is false.

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That's correct. – Myself Mar 9 '11 at 1:38

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The first is true. You are correct. For the second, truth would imply that 2x+1=1 has a positive integer as a solution, which is false, so you are correct again.

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It's worse than that for the second one, it would imply that for some $x\in\mathbf Z^+$ it holds that $2x-y = 1 = 2x-y'$ for all $y,y'\in\mathbf Z^+$, so it would imply that $y=y'$, thus $|\mathbf Z^+|=1$. – Myself Mar 9 '11 at 1:49

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