A property of a Set related to its Cardinality.

Countable

• Finite sets are countable
• The set of Positive Integers / Natural Numbers is countable
• Any infinite set with same cardinality as $\mathbb{N}$ is countable
• For countable infinite $S$, $|S|={\aleph }_0$ (aleph-null)

Hilbert’s Grand Hotel

Finite hotel: if all rooms occupied, new guest requires eviction.

Hilbert’s Grand Hotel: countably infinite rooms, all occupied — a new guest can still be accommodated (shift $n\to n+1$), showing $|\mathbb{N}|=|\mathbb{N}\cup \{x\}|$ without eviction.

Cantor’s Diagonalization

Technique proving $\mathbb{Q}$ is countable (diagonal enumeration), and $(0,1)\subset \mathbb{R}$ is uncountable — hence $\mathbb{R}$ is uncountable.

Uncountable

$\mathbb{R}$ is uncountable. Hint: show $(0,1)$ uncountable via diagonal argument, so any superset is uncountable.

Equality and Comparison

Two sets $A,B$ have same cardinality iff $\exists$ bijection $f:A\to B$ ($|A|=|B|$). If $\exists$ Injective $f:A\to B$ then $|A|\le |B|$; if injective but no bijection then $|A|<|B|$.