# Set Equivalence behaves like Equivalence Relation

## Theorem

Set equivalence behaves like an equivalence relation.

That is:

 $\ds \forall S:$ $\ds S \sim S$ Reflexivity $\ds \forall S, T:$ $\ds S \sim T \implies T \sim S$ Symmetry $\ds \forall S_1, S_2, S_3:$ $\ds S_1 \sim S_2 \land S_2 \sim S_3 \implies S_1 \sim S_3$ Transitivity

where $S, T, S_1, S_2, S_3$ are sets.

## Proof

For two sets to be equivalent, there needs to exist a bijection between them.

In the following, let $\phi$, $\phi_1$ and $\phi_2$ be understood to be bijections.

### Reflexive

From Identity Mapping is Bijection, the identity mapping $I_S: S \to S$ is a bijection from $S$ to $S$.

Thus there exists a bijection from $S$ to itself

Hence by definition $S$ is therefore equivalent to itself.

Thus $\sim$ is seen to behave like a reflexive relation.

$\Box$

### Symmetric

 $\ds$  $\ds S \sim T$ $\ds$ $\leadsto$ $\ds \exists \phi: S \to T$ Definition of Set Equivalence, where $\phi$ is a bijection $\ds$ $\leadsto$ $\ds \exists \phi^{-1}: T \to S$ Bijection iff Inverse is Bijection $\ds$ $\leadsto$ $\ds T \sim S$ Definition of Set Equivalence: $\phi^{-1}$ is also a bijection

Thus $\sim$ is seen to behave like a symmetric relation.

$\Box$

### Transitive

 $\ds$  $\ds S_1 \sim S_2 \land S_2 \sim S_3$ $\ds$ $\leadsto$ $\ds \exists \phi_1: S_1 \to S_2 \land \exists \phi_2: S_2 \to S_3$ Definition of Set Equivalence: $\phi_1$ and $\phi_2$ are bijections $\ds$ $\leadsto$ $\ds \exists \phi_2 \circ \phi_1: S_1 \to S_3$ Composite of Bijections is Bijection: $\phi_2 \circ \phi_1$ is a bijection $\ds$ $\leadsto$ $\ds S_1 \sim S_3$ Definition of Set Equivalence

Thus $\sim$ is seen to behave like a transitive relation.

$\blacksquare$

## Warning

It has been shown that set equivalence exhibits the same properties as an equivalence relation.

However, it is important to note that set equivalence is not strictly speaking a relation.

This is because the collection of all sets is itself specifically not a set, but a class.

Hence it is incorrect to refer to $\sim$ as an equivalence relation, although it is useful to be able to consider it as behaving like an equivalence relation.

## Also see

The definition of a cardinal of a set as the equivalence class of that set under set equivalence.