Negation of "For all x ∈ R, if ⌊5x⌋ = 5⌊x⌋, then x ∈ Z"

Question

Negation of "For all x ∈ R, if ⌊5x⌋ = 5⌊x⌋, then x ∈ Z"

Is there only one correct negation?

I know "There exists x ∈ R, so that ⌊5x⌋ = 5⌊x⌋, but x ∉ Z" is correct.

But what about: "For some x ∈ R, ⌊5x⌋ = 5⌊x⌋ and x ∉ Z" ?

Answer 1

What you wrote is equivalent. Here is another equivalency, but a shorter one: $$ \exists x \in \mathbb{R} \backslash \mathbb{Z} \text{ such that } \lfloor 5x \rfloor = 5\lfloor x\rfloor $$

Answer 2

Formally negation for $$(\forall x \in R)(⌊5x⌋ = 5⌊x⌋ \Rightarrow x \in Z)$$ is $$(\exists x \in R)(⌊5x⌋ = 5⌊x⌋ \land x \notin Z )$$ You can use several variants how to say it, but while you mean what is written, then they are same.