Proving a set is Uncountable or Countable Using Cantor's Diagonalization Proof Method

Question

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...

Answer 1

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.

Answer 2

Hints: 1) You just listed these numbers, so.... 2) Use the exact same technique of diagnolization you say you understand.

Answer 3

Set 1 is very countable; it can be placed into 1-1 correspondence with the integers. Set 2 is not countable. It is extremally disconnected and its intersection with $[0,1]$ is homeomorphic to a Cantor set.

Answer 4

For Set $2$: Just consider the subset of this set consisting of numbers in the interval $(0, 1)$. Assume you have a complete listing of such numbers, and write them out, padding them with trailing zeros as needed. (Remember that the defining characteristic of Set $2$ is that the numbers can be represented only with $2$s and $3$s—not that they cannot be represented otherwise.) Now each position in each number in the list is either a $0$, a $2$, or a $3$. If you can generate a number whose value in Set $2$ that nevertheless differs from the $n$th item in the list in the $n$th place value, then you have properly executed the diagonalization argument.

Since Set $2$ is a superset of its restriction to $(0, 1)$, if that restriction is uncountable, then Set $2$ itself must also be uncountable.