I'm trying to understand different sizes of infinities and what everything means (trying to piece everything together).
So if we find a bijection between a set $A$ and another set $B$ where $B$ is countable (countably infinite), then that implies $A$ is countable (is that correct).
Moreover, we can find a bijection between any two countable sets (I think this is correct).
If we find a bijection between two finite sets, then the two sets must be of the same cardinality.
A proper subset of a finite set, A has smaller cardinality than the set $A$.
However, every infinite subset of a countable set is countable. For example, $\mathbb{Z} \subset \mathbb{Q}$ and $\mathbb{Q}$ is countable and $\mathbb{Z}$ is countable.
I get confused, however, when dealing with uncountable sets.
Is it true that I can find always bijection between uncountable sets? For example, is it always possible to construct a bijection between two uncountable sets. Are there examples of cases where you can construct a bijection between two uncountable sets and cases where you cannot? For example, is there a bijection between $[0,1]$ and $\mathbb{R}$? (Can't think of one).
In general how do I prove a set is uncountable? I know that if it contains a set that is uncountable then it must be uncountable. Also, I have read through some diagonalization arguments although I find them a bit confusing. Is diagonalization the only way to prove uncountability? Or are there any other ways? What are examples of uncountable sets: $\mathbb{R}$, the set of binary sequences (I think, not sure), intervals such $[0,1]$, any other standard ones? (I want to practice proving sets are uncountable, so any suggestions on examples where I can prove a set is uncountable are much appreciated).If we find a bijection, between a set A and an uncountable set (given) does that mean A is uncountable as well?
Also to show two sets that are at most countable (either finite or countable) do not have the same cardinality do I show that it is impossible to construct a bijection between them.
Thanks.
Best Answer
Try to construct a bijection from (0,1) to R by shifting/scaling tangent function.