NL-NWO 1966 Problem 5

The image that maps $x$ to $1 - x$ is called complement, the image that maps $x$ to $\frac{1}{x}$ is called invert. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of complementationand/or inversion. A family is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family. Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$ -number family.