Problem #4: Injections and surjections on finite sets
In our previous problem we were exposed to the concept of Dedekind-finiteness, which suggests that finiteness can be defined in terms of injections and surjections, without reference to cardinal numbers.
Indeed, one of the most useful properties of finite sets combinatorially (here by finite we mean has n elements for some natural n) is that the injections and surjections from a given finite set to itself are the same.
Problem: Show that all injections from some canonical n-element set to itself are surjections, and vice versa.
Written on May 31, 2017