Countability

A set $A$ is countable iff $\,\exists f:A\rightarrow Z^{+}$, where $f$ is a one-one function.

Examples

• Countable: Any finite set, $\mathbb{Z}, \mathbb{Q}$
• Uncountable: $\mathbb{R}$, Any open/closed intervals in $\mathbb{R}$.

Transitive property

Say $B \subset A$. $$

$A \text{ is countable }\implies B \text{ is countable}$

$B \text{ is not countable }\implies A \text{ is not countable}$