Are the following sets countable or uncountable

Question

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.

Answer 1

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).

Answer 2

$a)$ countable Because the only freedom we have is the number of digits before the comma and the number of digits (possibly infinite) after the comma.

$b)$ Even the reals in the interval $[0,1]$ with $2$'s and $5$'s build an uncountable set which you can proof with Cantor's diagonal argument. (We have an infinite sequence with two possible values).

Answer 3

hint for the second:, the question could have been $1$ and $0$ instead, which is to say every number with binary representation.

For the first one, there is a bijection between those after the decimal and $\mathbb N$, and those before with $\mathbb N$ as well. From this, it's not hard to find something that works for $\mathbb N \times \mathbb N$.

Answer 4

As suggested by Peter, Cantor's diagonal argument is the way to do it for (b). For (a), however, you may simply note that any such number is rational.