Recent content by Mathmellow

I am trying to prove that any subset of a countable set is either finite or countable. I know that a set $$S$$ is countable if there exists a bijection that takes S to $$\Bbb{N}$$. My first thought was to consider the subset $$V$$ of $$S$$. If $$V$$ is finite we are done, since we can always...

I will definitely keep this in mind, I have a course in formal logic that I am planning to take so it will probably come in handy. Thank you for taking your time to respond to my questions! :D

Yes, of course. Thank you for the clarification though. This just became so much clearer, I really did not understand what the point of $j^*$ really was. So if I understand you correctly, the proof would have been simpler if they had used this statement? Was it a mistake the author made...

I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with. Theorem. Let m be a natural number. Then the...