I'm studying discrete maths, and currently on the topic of cardinality of sets.
I thought I got a basic grasp of countability, but I got this one wrong…
Determine whether the set of real numbers containing only a finite number of 1s in their decimal representation is countable or uncountable.
I thought it would be countable, or countably infinite rather.
As you could have…
1,1.1,11,1.11,11.1,111,…
…where you always start with a 1 before the decimal point, then shift whatever ones are to the right of the point up one, until there are no 1s left after the decimal point, then you restart with an extra one, and so forth.
It is apparently uncountable, but why? Have I interpreted the question wrong?
Best Answer
Hint: try counting the number constructed using only the digits $2$ and $3$ ... these contain zero ones (a finite number) [biject to binary expressions]