Fun with Fixed Points
TITLE: Fun with Fixed Points
Given a function f from a set to itself, a fixed point of f is just what you should expect: a point that doesn't move. To be more precise, it is a point x such that f(x)=x
Fixed points occur in many different settings. I'll discuss some of the ones that I find interesting.
Here is an easy but fun problem that you can certainly work out yourself. Let X be a finite set and let f be a function from X to itself which is chosen randomly from all such functions. Show that the probability that f has at least one fixed point is more than 63.2% (i.e., 79 out of 125).