The next key idea of Georg Cantor (1845 - 1918) was to use a given set $A$ as a representative for all sets, which are equipotent to it. In order to be able to do so, Cantor realized that any given set can be selected as a representative of a whole class of sets which are equipotent to it. This abstraction process allowed Cantor to ignore which elements a set has and to concentrate on how many elements it has.
Let $\mathcal X$ be a universal set. Being equipotent for any two sets \(A,B\subseteq \mathcal X\) is an equivalence relation \(\sim\) on \(\mathcal X\). Each equivalence class $[A]\in \mathcal X/\sim$1 is called the cardinal number or cardinality of $A$.
Instead of $[A]$, the notation $|A|$ is more commonly used for the cardinality of $A.$ Note that all cardinalities are disjoint.