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.
- integers not divisible by $3$
- integers divisible by $5$ but not by $7$
- the real numbers with decimal representations consisting of all $1s$
- the real numbers with decimal representations of all $1s\: \text{or}\: 9s$
