# Set Equivalence is Equivalence Relation

## Theorem

Set equivalence is an equivalence relation.

## Proof

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

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

### Reflexive

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

Thus there exists a bijection from $S$ to itself and $S$ is therefore equivalent to itself.

Therefore set equivalence is reflexive.

$\Box$

### Symmetric

\(\displaystyle \) | \(\) | \(\displaystyle S \sim T\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \exists \phi: S \to T\) | $\quad$ Definition of Set Equivalence | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \exists \phi^{-1}: T \to S\) | $\quad$ Bijection iff Inverse is Bijection | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle T \sim S\) | $\quad$ Definition of Set Equivalence: $\phi^{-1}$ is also a bijection | $\quad$ |

Therefore set equivalence is symmetric.

$\Box$

### Transitive

\(\displaystyle \) | \(\) | \(\displaystyle S_1 \sim S_2 \land S_2 \sim S_3\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \exists \phi_1: S_1 \to S_2 \land \exists \phi_2: S_2 \to S_3\) | $\quad$ Definition of Set Equivalence | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \exists \phi_2 \circ \phi_1: S_1 \to S_3\) | $\quad$ Composite of Bijections is Bijection: $\phi_2 \circ \phi_1$ is a bijection | $\quad$ | |||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle S_1 \sim S_3\) | $\quad$ Definition of Set Equivalence | $\quad$ |

Therefore set equivalence is transitive.

$\blacksquare$

## 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): $\S 17$: Theorem $17.1$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2.3$: Equivalence of sets (footnote $6$) - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.2$ - 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 $2.1$