Formal statement that functions intersect












0












$begingroup$


How do I express the following as a mathematical statement with quantification of all variables and making the universe explicit.



"The curves $y=1-x^2$ and $y=3x-2$ intersect"



So far I have $(exists (x,y)in mathbb{R})$






calculus proof-verification proof-explanation







share|cite|improve this question











$endgroup$












  • $begingroup$
    How would you say that a point $(x,y)$ is on one of the curves?
    $endgroup$
    – John Douma
    Jan 10 at 23:26










  • $begingroup$
    Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
    $endgroup$
    – Matteo
    Jan 10 at 23:28










  • $begingroup$
    Ahh right, if both are true. under the original condition
    $endgroup$
    – Forextrader
    Jan 10 at 23:29
















0












$begingroup$


How do I express the following as a mathematical statement with quantification of all variables and making the universe explicit.



"The curves $y=1-x^2$ and $y=3x-2$ intersect"



So far I have $(exists (x,y)in mathbb{R})$






calculus proof-verification proof-explanation







share|cite|improve this question











$endgroup$












  • $begingroup$
    How would you say that a point $(x,y)$ is on one of the curves?
    $endgroup$
    – John Douma
    Jan 10 at 23:26










  • $begingroup$
    Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
    $endgroup$
    – Matteo
    Jan 10 at 23:28










  • $begingroup$
    Ahh right, if both are true. under the original condition
    $endgroup$
    – Forextrader
    Jan 10 at 23:29














0












0








0





$begingroup$


How do I express the following as a mathematical statement with quantification of all variables and making the universe explicit.



"The curves $y=1-x^2$ and $y=3x-2$ intersect"



So far I have $(exists (x,y)in mathbb{R})$






calculus proof-verification proof-explanation







share|cite|improve this question











$endgroup$




How do I express the following as a mathematical statement with quantification of all variables and making the universe explicit.



"The curves $y=1-x^2$ and $y=3x-2$ intersect"



So far I have $(exists (x,y)in mathbb{R})$






calculus proof-verification proof-explanation




calculus proof-verification proof-explanation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 10 at 23:42









David G. Stork

11k41432




11k41432










asked Jan 10 at 23:20









ForextraderForextrader

808




808












  • $begingroup$
    How would you say that a point $(x,y)$ is on one of the curves?
    $endgroup$
    – John Douma
    Jan 10 at 23:26










  • $begingroup$
    Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
    $endgroup$
    – Matteo
    Jan 10 at 23:28










  • $begingroup$
    Ahh right, if both are true. under the original condition
    $endgroup$
    – Forextrader
    Jan 10 at 23:29


















  • $begingroup$
    How would you say that a point $(x,y)$ is on one of the curves?
    $endgroup$
    – John Douma
    Jan 10 at 23:26










  • $begingroup$
    Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
    $endgroup$
    – Matteo
    Jan 10 at 23:28










  • $begingroup$
    Ahh right, if both are true. under the original condition
    $endgroup$
    – Forextrader
    Jan 10 at 23:29
















$begingroup$
How would you say that a point $(x,y)$ is on one of the curves?
$endgroup$
– John Douma
Jan 10 at 23:26




$begingroup$
How would you say that a point $(x,y)$ is on one of the curves?
$endgroup$
– John Douma
Jan 10 at 23:26












$begingroup$
Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
$endgroup$
– Matteo
Jan 10 at 23:28




$begingroup$
Hint: if a point belongs to both lines (i.e. they intersect), then it's abscissa will give the same ordinate when substituted into the lines' equations.
$endgroup$
– Matteo
Jan 10 at 23:28












$begingroup$
Ahh right, if both are true. under the original condition
$endgroup$
– Forextrader
Jan 10 at 23:29




$begingroup$
Ahh right, if both are true. under the original condition
$endgroup$
– Forextrader
Jan 10 at 23:29










2 Answers
2






active

oldest

votes


















3












$begingroup$

No need to even mention the irrelevant $y$:



$$exists x in mathbb{R} s.t. 1-x^2 = 3 x-2$$



Just for "culture," here is a graph confirming there are two solutions:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you, that makes sense
    $endgroup$
    – Forextrader
    Jan 10 at 23:46



















1












$begingroup$

$$exists xin mathbb{R},, exists y in mathbb{R},,,y=1-x^2 ,wedge, y=3x-2$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
    $endgroup$
    – Forextrader
    Jan 10 at 23:30












  • $begingroup$
    Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
    $endgroup$
    – Matteo
    Jan 10 at 23:31











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

No need to even mention the irrelevant $y$:



$$exists x in mathbb{R} s.t. 1-x^2 = 3 x-2$$



Just for "culture," here is a graph confirming there are two solutions:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you, that makes sense
    $endgroup$
    – Forextrader
    Jan 10 at 23:46
















3












$begingroup$

No need to even mention the irrelevant $y$:



$$exists x in mathbb{R} s.t. 1-x^2 = 3 x-2$$



Just for "culture," here is a graph confirming there are two solutions:



enter image description here






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you, that makes sense
    $endgroup$
    – Forextrader
    Jan 10 at 23:46














3












3








3





$begingroup$

No need to even mention the irrelevant $y$:



$$exists x in mathbb{R} s.t. 1-x^2 = 3 x-2$$



Just for "culture," here is a graph confirming there are two solutions:



enter image description here






share|cite|improve this answer











$endgroup$



No need to even mention the irrelevant $y$:



$$exists x in mathbb{R} s.t. 1-x^2 = 3 x-2$$



Just for "culture," here is a graph confirming there are two solutions:



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 11 at 0:18

























answered Jan 10 at 23:43









David G. StorkDavid G. Stork

11k41432




11k41432












  • $begingroup$
    Thank you, that makes sense
    $endgroup$
    – Forextrader
    Jan 10 at 23:46


















  • $begingroup$
    Thank you, that makes sense
    $endgroup$
    – Forextrader
    Jan 10 at 23:46
















$begingroup$
Thank you, that makes sense
$endgroup$
– Forextrader
Jan 10 at 23:46




$begingroup$
Thank you, that makes sense
$endgroup$
– Forextrader
Jan 10 at 23:46











1












$begingroup$

$$exists xin mathbb{R},, exists y in mathbb{R},,,y=1-x^2 ,wedge, y=3x-2$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
    $endgroup$
    – Forextrader
    Jan 10 at 23:30












  • $begingroup$
    Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
    $endgroup$
    – Matteo
    Jan 10 at 23:31
















1












$begingroup$

$$exists xin mathbb{R},, exists y in mathbb{R},,,y=1-x^2 ,wedge, y=3x-2$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
    $endgroup$
    – Forextrader
    Jan 10 at 23:30












  • $begingroup$
    Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
    $endgroup$
    – Matteo
    Jan 10 at 23:31














1












1








1





$begingroup$

$$exists xin mathbb{R},, exists y in mathbb{R},,,y=1-x^2 ,wedge, y=3x-2$$






share|cite|improve this answer









$endgroup$



$$exists xin mathbb{R},, exists y in mathbb{R},,,y=1-x^2 ,wedge, y=3x-2$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 10 at 23:26









MindlackMindlack

4,830210




4,830210












  • $begingroup$
    Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
    $endgroup$
    – Forextrader
    Jan 10 at 23:30












  • $begingroup$
    Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
    $endgroup$
    – Matteo
    Jan 10 at 23:31


















  • $begingroup$
    Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
    $endgroup$
    – Forextrader
    Jan 10 at 23:30












  • $begingroup$
    Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
    $endgroup$
    – Matteo
    Jan 10 at 23:31
















$begingroup$
Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
$endgroup$
– Forextrader
Jan 10 at 23:30






$begingroup$
Thank you, but wouldn't this also suffice $(exists (x,y)in mathbb{R},y=1-x^2 wedge y=3x-2$ )
$endgroup$
– Forextrader
Jan 10 at 23:30














$begingroup$
Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
$endgroup$
– Matteo
Jan 10 at 23:31




$begingroup$
Isn't this statement redundant? It's sufficient to state that there exists an $x$ for which $1-x^2=3x-2$. Correct?
$endgroup$
– Matteo
Jan 10 at 23:31


















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