I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list.
I have to show whether these sets are countable or uncountable. If countable, how would you enumerate the set? If uncountable, how would you prove using diagonalization?
Set 1. All real numbers represented only by 1's. EX) 1, .11, 111.11, 1.111…
Set 2. All real numbers represented only by 2's and 3's. EX) .2, 23.2, 22.2232…
Best Answer
Set $2$ can be put into one-to-one correspondence with the binary representation of the reals by the map that takes $2$ to $0$ and $3$ to $1$. Thus, this set has the same cardinality as $\mathbb R$ which is uncountable.