Write symbolically: negation and useful denial

[ver_mathstats]

Homework Statement

Express the statement symbolically, including a quantification of all variables which makes the universe explicit. Negate the symbolic statement, and express the negation in natural language as a useful denial.

The curves y = 1−x² and y = 3x-2 intersect.

2. Homework Equations

The Attempt at a Solution

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I am unsure of how to write this statement symbolically.

I assume I start with (∃x∈U) since there is only two point of intersections.

However I do not know where to go from there.

Would I have to equate them to each other?

Would doing 1−x² = 3x-2 be correct?

Thank you.

 

[fresh_42]

I would start to write the curves as sets and consider pairs $(x,y)\in U$ not only $x$:
$P:=\{\,(x,y)\,:\,y+x^2-1=0\,\}$ and $S:=\{\,(x,y)\,:\,y-3x+2=0\,\}$. This way you can handle the intersection as intersection of sets.

 

[ver_mathstats]

I would start to write the curves as sets and consider pairs $(x,y)\in U$ not only $x$:
$P:=\{\,(x,y)\,:\,y+x^2-1=0\,\}$ and $S:=\{\,(x,y)\,:\,y-3x+2=0\,\}$. This way you can handle the intersection as intersection of sets.

Thank you for the help, I have not yet learned how to work in sets but I will try to complete my answer using this way. But could this question also be completed if they were to be equated to each other?

 

[fresh_42]

Thank you for the help, I have not yet learned how to work in sets but I will try to complete my answer using this way. But could this question also be completed if they were to be equated to each other?

You can solve it this way, but a solution is not requested, only a formal statement about the set of solutions. The statement does not have to be true, but unfortunately they do intersect. But again, a statement about a solution is something else than the actual solution. So even if you write it as $x^2+3x-3=0$ then you still have to say something about the solution of this quadratic equation.