Functions

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