Definition:Set Equivalence

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Definition

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 and only 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:


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:

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


Examples

Arbitrary Example $1$

Let:

\(\ds S\) \(=\) \(\ds \set {2, 4, 6, 8}\)
\(\ds T\) \(=\) \(\ds \set {3, 5, 7, 9}\)

There exists a bijection between $S$ and $T$, that is: $f : x \mapsto x + 1$, for example.

They both have the same cardinality, that is $4$.


Arbitrary Example $2$

Let:

\(\ds A\) \(=\) \(\ds \set {1, 2, 3}\)
\(\ds B\) \(=\) \(\ds \set {1, 2, 3, 4, 5}\)

There does not exists a bijection between $A$ and $B$.

That is because they do not have the same cardinality.


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.


Sources