Definition:Cardinality/Also known as

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Some authors prefer the term order instead of cardinality, particularly in the context of finite sets.

Georg Cantor used the term power and equated it with the term cardinal number, using the notation $\overline {\overline M}$ for the cardinality of $M$.

Some sources cut through all the complicated language and call it the size.

Some sources use $\map {\#} S$ (or a variant) to denote set cardinality. This notation has its advantages in certain contexts, and is used on occasion on this website.

Others use $\map C S$, but this is easy to confuse with other uses of the same or similar notation.

A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$.

1968: A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis use $\map m A$ for the power of the set $A$.

Further notations are $\map n A$ and $\overline A$.