Determine whether each of these sets is countable or uncount

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 Is the real numbers with decimal representations of all Is or 9s

Solution

(a). integers not divisible by 3.: Countable.: As we know that a set of positive integers is countable, and our set is a collection of positive integers minus, the set of integers which are divisible by 3 i.e., the set 0, 3, 6, ...

(b). integers divisible by 5 but not by 7.: Countable. As we know that a set of positive integers is countable, and our set is a collection of positive integers minus, the set of integers which are divisible by 5 but not by 7 i.e., the set 5, 10, 15, 20, 25, 30, 40, 45, ...

(c). the real numbers with decimal representations consisting of all 1s.: Uncountable.

(d). the real numbers with decimal representations of all 1s or 9s.: Uncountable.