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I've asked a question similar to this like an hour ago. However, I still don't clearly grasp what I should do to come to the answer.... The instructions are as follow:

Write the following statements in predicate form, using logical operators $\wedge$,$\vee$,$\neg$, and the quantifiers $\forall$, $\exists$. Below $\mathbb Z^+$ denotes all positive integers $1,2,3,...$

My statement is as follow: the equation $x_3+y_3=z_3$ has no solution $x,y,z\in \mathbb Z^+$

my attempt is pretty poor at it. I thought of maybe describing the function as $P(x,y)$. Knowing that if $\neg P(x,y)$ is true, then there are no solutions. So

$$\forall x\forall y( \neg P(x) \wedge z \in \mathbb Z^+)$$

I don't know if that makes sense :/

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  • $\begingroup$ Do you mean the equation $x^3+y^3=z^3$? $\endgroup$ – russoo Feb 4 '15 at 2:39
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$$\forall x \forall y ( \neg P(x) ∧ z \in \Bbb Z^+)$$

Close. You seem to be trying to say that: for every pair of positive integers, $x, y$, there does not exist a positive integer $z$ such that $z^3=x^3+y^3$.

$$ \forall x\,\forall y\;\neg \exists z\; \Big(\big((x\in \Bbb Z^+)\wedge (y\in \Bbb Z^+)\big)\;\to\; \big((z\in \Bbb Z^+)\wedge (z^3=x^3+y^3)\big)\Big) $$

In shortened form

$$ \forall x\in \Bbb Z^+,\forall y\in \Bbb Z^+, \neg \exists z\in \Bbb Z^+ : (z^3=x^3+y^3) $$


Personally I'd just say: there's no triple of positive integers $x,y,z$ such that $z^3=x^3+y^3$.

$$\forall (x,y,z)\in {\Bbb Z^+}^3 (z^3\neq x^3+y^3)$$

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  • $\begingroup$ Maybe I'm missing something. I think I would just write $(\forall x,y,z\in\mathbb{Z^+})(x^3+y^3\neq z^3)$ and move on to the next problem. $\endgroup$ – Daniel W. Farlow Feb 4 '15 at 3:18
  • $\begingroup$ Well I was thinking that because there does not exist ANY that I'd have to use a universal quantifier no? saying there exist no x,y,z that satisfies the function, right? $\endgroup$ – Marc-Andre Leclair Feb 4 '15 at 3:21

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