Answer to Question #175943 in Real Analysis for Nikhil Singh

Question

Check whether the intervals [2,5] and [7,10] are equivalent or not

Answer 1

Consider the function $f:[2,5]\to[7,10],\ f(x)=x+5.$ For each $y\in[7,10]$ the equation $f(x)=y$ (which is equivalent to $x+5=y$) has a unique solution $x=y-5\in[2,5]$. Therefore, the function $f$ is a bijection. It follows that the intervals $[2,5]$ and $[7,10]$ are equivalent.