Injective if and only if Surjective
Hi there, I have this problem and while I think what I've put is correct I'm not sure that it is rigorous enough. Any help much appreciated!!
(a) Let A be a finite set. Show that a function f : A → A is injective if
and only if it is surjective.
(b) Show that this result is false for infinite sets by exhibiting
• An injective map f : N → N that is not surjective;
• A surjective map f : N → N that is not injective.
For a) I've put that a function is injective if each unique element x in the domain X is mapped to an element y in some co-domain Y, and a function is surjective if for every element y in the co-domain Y there is atleast one element x in the domain X such that f(x)=y. As the cardinality of the domain and co-domain are the same if the function is injective it must also be surjective.
For b) An injective map which is not surjective would be 2x=y
For a surjective map which is not injective I'm thinking something like ┌ x/2┐=y (those symbols are meant to denote the ceiling function!!)