Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
- all bit strings not containing the bit $0$
- all positive rational numbers that cannot be written with denominators less than $4$
- the real numbers not containing $0$ in their decimal representation
- the real numbers containing only a finite number of $1s$ in their decimal representation
