14. Statement: If X is a finite set and f is an injective function from X→X, then f is bijective.

Proof: Let ** f** be an injective function from a set X to a set Y, both with cardinality n. The statement is vacuously true for n=0. We can then assume that for any two sets that each have equal cardinality less than some positive integer n, any injection between them is also surjective. Now remove the point (

*a, f(a)*) from the set. By injection,

*f(a)*has only the preimagine

*a*and by virtue of being a function,

*a*only has the imagine

*f(a)*; removing this pair removes

*a*from the domain and

*f(a)*from the codomain. The resulting function is a set with cardinality n-1, and by the assumption, it is assumed to be surjective. Any function that is both injective and surjective is bijective.

Comments - Andrew Furash

Great work!