Let S and T be two countably infinite sets. Suppose S is not a subset of T and vice-versa. Is the intersection of S and T a finite set? If not, please provide counterexamples.
I originally asked this question because it occurred to me when trying to prove that the union of two countably infinite sets is countable. I recently picked up Ralph Boas' Primer of Real Functions and have been trying to do the exercises. However, in the book, Boas doesn't introduce the notion of countability using injective functions and I have been trying to come up with a proof that doesn't involve injections. I fooled myself with a 'proof' but now I'm just stuck. Any hints to help me along the way?
I'm very new to real analysis. So please bear with me.
Also, what proportion of exercises should I be able to do if I really understand the material?
I guess what I'm asking is that how do the exercises serve as a barometer for my mathematical skill?
Best Answer
Not necessarily. Take $S=\{1\}\cup\{3,4,5,\ldots\}$. $T=\{2\}\cup\{3,4,5,\ldots\}$.
Perhaps more interesting:
$S=\{-1,-3,-5,\ldots\}\cup\{1,2,3,\ldots\}$ and $T=\{-2,-4,-6,\ldots\}\cup\{1,2,3,\ldots\}$