# Cardinals form Equivalence Classes

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## Theorem

Let $\map \Card S$ denote the cardinal of the set $S$.

Then $\map \Card S$ induces an equivalence class which contains all sets which have the same cardinality as $S$.

## Proof

Follows directly from:

- The definition of a cardinal as $S \sim T \iff \map \Card S = \map \Card T$
- Set Equivalence behaves like Equivalence Relation
- Relation Partitions Set iff Equivalence.

$\blacksquare$