Calculate cardinal number of given A set

Question

Let $x \in \mathbb R$.

Let $\text A x = \{ y \in \mathbb R : |x-y| \in \mathbb Z \}$

Find $| \text A x |$.

Now I understand I need to use a bijective function, to send $\text A x$ to a known caridnal number group like $\mathbb N$ which is $\aleph_0$.

is that the correct approach? a bit stuck.

Answer 1

x=[x]+{x},y=[y]+{y}. |x-y| is in Z iff {x}={y}.
Therefore elements of S=Ax are uniquely determined by [y]. Thus |S|=Z=N