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 :/
$$\forall x \forall y ( \neg P(x) ∧ z \in \Bbb Z^+)$$
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 $z^3=x^3+y^3$.
$$\forall (x,y,z)\in {\Bbb Z^+}^3 (z^3\neq x^3+y^3)$$