# Give a useful written negation of the statement

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 $y = e^z$.

(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$$R^+) [(\existsx,z\in$$R$)($R$$\wedge$$S$)]

(ii) ($\exists$y$\in$$Uy)[(\forallx\in$$U$x)$\neg$$R(x)\vee(\forallz\in$$U$z)$\neg$$S(z)] (iii) There is a positive real number y such that for all real numbers x, y, y \neq x^2-1 or y \neq e^z I am not quite sure on the negation though. • Your answer is correct. It’s nicer to show the parameters of R and S, though: R(x,y): y=x^2-1, for example. (Your P and Q aren’t usually called statements. The negation is$$\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. – Steve Kass Oct 2 '14 at 5:02