Cardinality

The cardinality¹ of a set is a Cardinal uniquely corresponding to the set's Isomorphism class within $𝖲 𝖾 𝗍$ . #m/def/set

$| 𝐴 | = | 𝐵 | ⟺ 𝐴 ≅ 𝐵$ iff there exists a bijection between sets $𝐴$ and $𝐵$ . $𝐴$ and $𝐵$ are thence said to be equinumerous.
$| 𝐴 | \leq | 𝐵 |$ if and only if there exists an injection $𝑓 : 𝐴 ↣ 𝐵$ , or equivalently iff $𝐴$ is equinumerous with some $𝐶 \subseteq 𝐵$ .

For finite sets, cardinality is given by the number of elements in the set. A set $𝐴$ with $| 𝐴 | \leq | ℕ | = ℵ 0$ is called countable. The naturals have the smallest transfinite cardinality.

Properties

Upper bound on the cardinality of an arbitrary union

#state/tidy | #lang/en | #SemBr

Mächtigkeit ↩

Cardinality

Properties

Footnotes