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