Mathematics Question from JEE Main 2023 · 6 April, Shift 2

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x) = \dfrac{1}{\sqrt{[x] - x}}$, where $[x]$ denotes the smallest integer greater than or equal to $x$. Then among the statements:

(S1): $A \cap B = (1, \infty) - \mathbb{N}$ and

(S2): $A \cup B = (1, \infty)$