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

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)]