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

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

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