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 $B \subset A$.

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

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