Naïve set theory MOC

Cardinality

The cardinality^[Mächtigkeit] of a set is a Cardinal uniquely corresponding to the set’s Isomorphism class within Category of sets. 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| ≤|𝐵| if and only if there exists an injection 𝑓 :𝐴 ↣𝐵, or equivalently iff 𝐴 is equinumerous with some 𝐶 ⊆𝐵.

For finite sets, cardinality is given by the number of elements in the set. A set 𝐴 with |𝐴| ≤|ℕ| =ℵ0 is called countable. The naturals have the smallest transfinite cardinality.

Properties

• Upper bound on the cardinality of an arbitrary union

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