The following statement had been proven. What I need help with is transforming it into its equivalent logical notation. Please check if my use of predicates and existential quantifier is correct.
There are integers $a$ and $b$ such that $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$.
Below is my attempt in writing the above statement in its logical notation: $$ \begin{align} P(x) &: x \in \mathbb{Z}\\ Q(x) &: x \in \mathbb{Z}\\ S(x,y) &: \sqrt{x+y}=\sqrt{x} + \sqrt{y} \end{align}$$
$\exists x \; \exists y \; (P(x) \wedge Q(y)) \wedge S(x,y)$
Is this correct?
My variant: $$(\exists x \in \mathbb{Z})(\exists y \in \mathbb{Z})(\sqrt{x + y} = \sqrt{x} + \sqrt{y})$$