real analysis - Are any two uncountable sets similar to each other?

Two sets $A$ and $B$ are called similar $\iff$ thee exists a one to one function $F$ whose domain is the set $A$ and whose range is the set $B$.

We know that two countably infinite sets should be similar to each other but what about uncountable sets?

Attempt: Let $a_1,a_2,a_3,\cdots$ be elements in $A$ and $b_1,b_2,b_3,\cdots$ in $B$.

Then, can we say that this mapping : $a_1 \rightarrow b_1, a_2 \rightarrow b_2, \cdots$ is a one one onto correspondence between $A$ and $B$? I am a little confused.

EDIT: My query actually arose from this problem : If $B$ is uncountable and $A$ is uncountable, then we need to prove that $B$ is similar to $B-A$. I proved that $B-A$ is uncountable and $B$ is already uncountable. Now, I should prove that $B$ and $B-A$ are similar. Could you please give me a hint?

Thank you for your help.

Answer

Uncountable just means not countable. Much like infinite means not finite, and there are many different ways to be not finite, there are different ways to be uncountable.

More specifically $\mathcal P(\Bbb R)$ is uncountable, and by Cantor's theorem it is not similar to $\Bbb R$.