# 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.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 13$: Arithmetic - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.7$. Similar sets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets: Theorem $17.1$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2.3$: Equivalence of sets (footnote $6$) - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.4$: Functions: Problem Set $\text{A}.4$: $26$ - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $2$. Definition of Equivalence. The Concept of Cardinality. The Axiom of Choice: Theorem $2.1$

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.2$