Represent the statement below into symbolic form (without any word) using quantifiers such that the $"\neg"$ symbol is not used at the beginning statement.
"Graph of functions $y=x^2+x+2$ and $y=x-2$ are not intersects".
How to do it? Should I define first the equation of functions? But I also got confusing. Any ideas?
Attempt: Let $P(x,y)$ be the statement "$y=x^2+x+2$" and $Q(x,y)$ be the statement "$y=x-2$". Let $\Bbb R$ be the universe. Then $$(\forall x)(\forall y)\overline{P(x,y) \wedge Q(x,y)}.$$
Or
Let $P(x,y)$ be the statement $"y=x^2+x+2=x-2"$. Let $\Bbb R$ be the universe. Then $$(\forall x,y)\overline{P(x,y)}.$$
Does these true?
Let f(x)=x^2+x+2 and g(x)=x-2. If you want to be less precise, let ‘H(x,y)’ be the predicate “x is a graphical point of intersection with y”. Then, we can define your sentence as:
∀x∀y(x,y∈ℝ—>~(H(f(x),g(y))v
H(g(y),f(x)))).
If you want to more precision, you can say:
∀x∀y(x,y,∈ℝ—>~((f(x)=y—>~(g(x)=y))—>
(H(f(x),g(y))vH(g(y),f(x)))))
If you don’t want to explicitly mention that the graphs don’t intersect, then you can simply say:
∀x∀y(x,y,∈ℝ—>(f(x)=y—>~(g(x)=y))).