# 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 $2.1$

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