Definition:Set Equivalence

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Let $S$ and $T$ be sets.

Then $S$ and $T$ are equivalent if and only if:

there exists a bijection $f: S \to T$ between the elements of $S$ and those of $T$.

That is, if they have the same cardinality.

This can be written $S \sim T$.

If $S$ and $T$ are not equivalent we write $S \nsim T$.

Also known as

Other terms that are used that mean the same things as equivalent are:

  • Equipotent (equalness of power), from which we refer to equivalent sets as having the same power
  • Equipollent (equalness of strength)
  • Equinumerous (equalness of number)
  • Similar.

Also denoted as

Some sources use $S \simeq T$ instead of $S \sim T$ to denote equivalence.

Other notations for $S \sim T$ include:

$S \mathrel {\operatorname {Eq} } T$
$\map {\mathrm {Eq} } {S, T}$

Also see

  • Results about set equivalence can be found here.

Historical Note

The notion of set equivalence was first introduced by Georg Cantor in $1878$.

The term he used was Mächtigkeit.