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"There is one and only one real solution to the equation x^3 + x + 1 = 0"

Could someone please explain to me how to write this using quantifier notation?

Three.OneFour
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PSA BITCOIN
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  • Start from the easy part : "there is (at least) one solution...". At least one is $\exists x$. – Mauro ALLEGRANZA Feb 06 '19 at 07:55
  • Are you familiar with (/ allowed to) use the quantifier ∃! , which means exactly "there exists one and only one"? If so, could you elaborate your problem? If not, the uniqueness can be rephrased as "any two objects with this property are equal". – Three.OneFour Feb 06 '19 at 13:48

3 Answers3

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enter image description here

This one might serve the purpose.

user642005
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    Welcome to math.se! You can write math formulas using LaTeX on this site if you enclose them with $ or $$. As for your answer, you're kind-of capturing the idea for uniqueness, but it doesn't make sense to enumerate the solutions. Your expression also doesn't guarantee existence. – Three.OneFour Feb 06 '19 at 14:10
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$\exists !x \in \mathbb {R} (x^3+x+1=0)$

In MathJax/TeX, "\exists !x \in \mathbb {R} (x^3+x+1=0)"

bjcolby15
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  • The $\exists !$ notation handles this, but it's possible that the OP is asking for a solution using $\forall$ and $\exists$. That's a relatively easy exercise. – Brian Borchers Feb 06 '19 at 01:47
  • Thanks - I haven't done quantifiers in ages...I've edited my response above. – bjcolby15 Feb 06 '19 at 12:52
  • I think you misunderstood the comment, your old solution was fine (the new one is not, you cannot have two quantifiers for one variable), but the point of the question may habe been to not use this special quantifier ∃! but only ∃ and ∀. – Three.OneFour Feb 06 '19 at 13:54
  • Yup...user642005 has a better solution below with the $\forall$ and $\exists$. – bjcolby15 Feb 06 '19 at 14:34
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If you can't use the $\exists !$, use:

$$\exists x \in \mathbb {R} \ \exists y \in \mathbb {R}(y^3+y+1=0 \leftrightarrow y = x)$$

Bram28
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