I am translating the negation of the statement $(\exists (x,y) \in R, y= 3x-2 \wedge y = 1-x^2) $
$(\forall (x,y) \in R, y\ne 3x-2 \vee y \ne 1-x^2) $
So far I have, "For every $(x,y)$ there exists a real number such that $y\ne 3x-2 $ or $ y \ne 1-x^2$ "
The reason I am second guessing myself (even if i am correct) is that it doesn't really make sense to me in the form of a sentence. Am I right? If not, any suggestions?
Note:
$$ \lnot (\forall x \in A, \phi(x) \wedge \psi(x))$$
is the same as
$$ \exists x \in A, (\lnot\phi(x)) \vee (\lnot \psi(x))$$
If $\wedge$ and $\vee$ swapped places in the above, you still get a true statement.
If $\exists$ and $\forall$ are swapped places in the above, you still get a true statement.
Your correct if you take out the "the exists part". That is, the correct statement is: "For every $(x,y), y \ne 3x - 2 \vee y \ne 1 - x^2$".
Note that in your case, $A = \mathbb{R}^2$, so elements in $A$ are of the form $(x,y)$ where $x \in \mathbb{R}$ and $y \in \mathbb{R}$.
You asked about readability. For the first sentence, you have "There exists a pair $(x,y)$ such Eq1 is true and Eq2 is true". The negation of this statement is that "There doesn't exist a pair $(x,y)$ such that Eq1 is true and Eq2 is true". That is, "For every pair $(x,y)$, either Eq1 is false or Eq2 is false".