A countable set is a set whose elements can either be counted one-by-one (infinite but countable) or are finite in number. Specifically, a set $S$ is called countable if either:

1. Finite Countable Set: The set has a finite number of elements, or
2. Infinite Countable Set: The set can be placed in a one-to-one correspondence with the set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$.

Formal Definition:

A set $S$ is countably infinite if there exists a bijection (one-to-one and onto function) between $S$ and the natural numbers $\mathbb{N}$. In other words, the elements of $S$ can be indexed by natural numbers.

Examples of Countable Sets:

1. Natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$.
2. Integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ (though infinite, they can be mapped to $\mathbb{N}$).
3. Rational numbers $\mathbb{Q} = \left\{\frac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0\right\}$ are countably infinite, though not obvious at first glance.
4. Finite sets like $\{1, 2, 3\}$ are trivially countable.

Uncountable Set:

If a set cannot be put into a one-to-one correspondence with $\mathbb{N}$, it is called uncountable. For example, the set of real numbers $\mathbb{R}$ is uncountable.

Key Insights:

• Every finite set is countable.
• Infinite sets may be either countable or uncountable.
• Countable sets can be listed or indexed by natural numbers.

Do you want more details or have any questions on this topic? Here are some related questions:

1. Why are rational numbers countable while real numbers are not?
2. How do you prove that the set of integers is countable?
3. What is the difference between countable and uncountable infinity?
4. Can the union of countably infinite sets be uncountable?
5. What is the Cantor diagonalization argument for showing that $\mathbb{R}$ is uncountable?

Tip: To understand countability better, practice creating a one-to-one correspondence between infinite sets and $\mathbb{N}$!