For any two sets A,B; there exists a bijection between A and B
I have read following two definitions of infinite sets:
1. Set S, is finite if for some . If a set is not finite, it is infinite.
2. Set is infinite if there exits a proper subset of S, say S' such that
Problem: I want to prove the equivalence of these two definitions.
My Attempt: I have to demonstrate both Part 1 and Part 2 below -
Part 1: To prove, if there exists no such that
Outline of my attempt
1. Let for some
2. Prove where
4. Now we have . This leads to a contradiction if you apply pigeon-hole principle. Hence assumption in step 3 is not valid and we are done.
Problem I am facing: Step 2 above. How do I prove it formally. I know it sounds so much true from intuition but how do I show it in the language of formal mathematics.
Part 2: To prove, if there doens't exist any such that there exists a proper subset of S, S', where
Problem I am facing: Don't even know how to start with this one?
Also is my approach to this question correct, or there are other ways to do it?