A set is countable if and only if each of its elements can be associated with a different positive integer. Every finite set is countable. For example, the set {2, 3, 5, 7, 11} is countable.

1↔2

2↔3

3↔5

4↔7

5↔11

Infinite sets can also be countable. For example, the set of all prime numbers is countable.

1↔2

2↔3

3↔5

4↔7

5↔11

6↔13

7↔17

8↔19

9↔23

10↔29

...

1. Is the set of all integers countable?

2. Is the set of all positive rational numbers countable?

3. Is the set of all rational numbers countable?

4. Is the set of all positive real numbers countable?

5. Is the set of all real numbers countable?

Part 4. No.

Cantor proved this with his (ingenious) "diagnoal argument." See https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Part 5. Also no, since a subset - the positive reals - is uncountable.

Extra credit: The reals are formed from the union of the rationals and the irrationals. Since the rationals are countable, the uncountability of the reals implies the uncountability of the irrationals, too.