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

Question

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

only $(\mathrm{S} 2)$ is true

only (S1) is true

neither (S1) nor (S2) is true

both (S1) and (S2) are true

Solution

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