# Proof

(related to Proposition: Cardinal Number)

It is sufficient to show that being equipotent is an equivalence relation $$"\sim"$$ on the sets belonging to a given (non-empty) set system $$\mathcal X$$. In the following, let $$A,B,C\subseteq\mathcal X.$$

• Since there is always a bijective function $$f:A\to A$$ (i.e. the set $$A$$ is equipotent to itself), we have that $$A\sim A$$. Therefore, "$\sim$" is reflexive.
• Let $$A\sim B$$. Since there is bijective function $$f:A\to B$$, it is invertible with $f^{-1}:B\to A.$ Thus, $B\sim A$ and "$\sim"" is symmetric. • Let $$A\sim B$$ and $$B\sim C$$. Therefore there are bijective maps $$f_1:A\to B$$ and $$f_2:B\to C$$. Since the composition of bijective functions is bijective, the is bijective function $$f:=f_2\circ f_1 :A\to C$$. Thus $$A\sim C$$, and "$\sim"" is transitive.
• Therefore, "$\sim$" is an equivalence relation, by definition.

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### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition