Let the sets \(A\) and \(B\) denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{[x]-x}}\), w…

Let the sets \(A\) and \(B\) denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{[x]-x}}\), where \([x]\) denotes the smallest integer greater than or equal to \(x\). Then among the statements \(\left(\mathrm{S}_1\right): A \cap B=(1, \infty)-N\) and \[
\left(\mathrm{S}_2\right): A \cup B=(1, \infty)
\]

[JEE Main 2023, 6 Apr (Shift 2)]