Definition:Set Equivalence

Definition
Let $S$ and $T$ be sets.

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

That is, 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 $S \simeq T$ or $S \approx T$ 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

 * Definition:Cardinality
 * Definition:Count


 * Set Equivalence behaves like Equivalence Relation