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I am struggling a bit with giving the useful negation and the written statement:

For (b) below:

(i) Assign a universal set to each variable, label each component statement with a symbol;

(ii) write a useful negation of the statement symbolically; and

(iii) Give a useful written negation of the statement.

(b) For every positive real number y there exist real numbers x and z such that $y = x^2 − 1$ and $y = e^z$.

Here is my answer:

(i) $U$x = $R$, $U$y = $R^+$, $U$z = $R$,

$R(x,y)$: $y = x^2 -1$,

$S(x,y)$: $y = e^z$

($\forall$y$\in$$R^+$) [($\exists$x,z$\in$$R$)($R$$\wedge$$S$)]

(ii) ($\exists$y$\in$$U$y)[($\forall$x$\in$$U$x)$\neg$$R$(x)$\vee$($\forall$z$\in$$U$z)$\neg$$S$(z)]

(iii) There is a positive real number $y$ such that for all real numbers $x$, $y$, $y$ $\neq$ $x^2-1$ or $y$ $\neq$ $e^z$

I am not quite sure on the negation though.

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    $\begingroup$ Your answer is correct. It’s nicer to show the parameters of $R$ and $S$, though: $R(x,y): y=x^2-1$, for example. (Your $P$ and $Q$ aren’t usually called statements. The negation is $$\exists y\in \mathbb{R}^+\mbox{ such that } \forall x,z\in\mathbb{R}, [\neg(R(x,y)\lor S(y,z))]$$ This can be written as $$\exists y\in \mathbb{R}^+\mbox{ such that } \forall x,z\in\mathbb{R}, [\neg R(x,y)\lor \neg S(y,z))]$$ and expressed in prose just as you did. $\endgroup$ – Steve Kass Oct 2 '14 at 5:02
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The difficulty would be only in express iterated negations of quantifiers in the statement, but you did a great job. Both your informal (iii) and formal (ii) negations are correct.

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  • $\begingroup$ @Carlos Anytime, keep your good work! $\endgroup$ – Bruno Bentzen Oct 2 '14 at 12:40

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