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 :/

  • $\begingroup$ Do you mean the equation $x^3+y^3=z^3$? $\endgroup$ – russoo Feb 4 '15 at 2:39

$$\forall x \forall y ( \neg P(x) ∧ z \in \Bbb Z^+)$$

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 $z^3=x^3+y^3$.

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

| cite | improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.