I have been trying to figure out this problem for a while now,
Determine whether the following sets are countable or uncountable. Prove your answer.
a)the set of real numbers with decimal representation consisting of all 2’s (2.22 and 22.222 . . . are such numbers).
b)the set of real numbers with decimal representation consisting of 2’s and 5’s
For a, i would say that it is countable due to that I can have a base of 2 to where I can count up for example 2.2, 2.22,… 222.22222 and so on.
For b, I was confused on how to even start this one.
I am just looking for a push in the right direction and if I am some what correct for the first one.
Best Answer
For a) you're right that it's countable. However, it's not immediately obvious how to list the elements, since there are infinitely many numbers in the set which lie between $2.2$ and $22.2$. You can think of the numbers as being described by two integers (how far to the left of the decimal point it extends, and how far to the right it extends). Do you know how to show that $\mathbb N\times\mathbb N$ is countable?
For b) you should try to construct a bijection with the real numbers, which will show it's uncountable (hint: binary).