Definition of floor in terms of <

Question

I have read a proof which includes the following step:

$\lnot(n \lfloor x \rfloor < \lfloor nx \rfloor) \equiv \lnot(n \lfloor x \rfloor < nx)$.

So the rule required by this step appears to be of the form:

$n < \lfloor x \rfloor \equiv n < x$.

My trouble is that I don't see how this follows from the definition I've been using:

$n \leq \lfloor x \rfloor \equiv n \leq x$.

Is the proof step valid?

Admittedly, I'm weak with inequalities, so please be gentle if I've missed something obvious!

Thanks

Answer 1

I do not think that that proofstep is valid. Set $n=2$ and $x=2.1$. Then the left hand side becomes

$\neg(4<4)=\text{TRUE}$

whereas the right hand side becomes

$\neg(4<4.2)=\text{FALSE}$

so the two cannot be equivalent.