Disprove why 0 ∉ Z

Question

Disprove this

A value x is said to be an integer when

floor(x) = x, where x ∈ ℝ

floor(x)/x = 1

Therefore

floor(x)/x ∈ Z, where x ∈ ℝ

And since 0 ∈ ℝ

From the definition of an integer,

floor(x)/x ∈ Z, where x ∈ ℝ

if 0 ∈ Z,

floor(0) = 0

Then

floor(0)/0 = 1 ∉ Z

Answer 1

There are two big flaws:

• $\frac00$ is not defined and there's nothing you can conclude about it.

• there is no justification of your final $\notin$. On the opposite, $1\in\mathbb Z$.

Answer 2

Your definition of "integer" is wrong - we do not require $\lfloor x\rfloor/x$ to equal $1$.