Let $X$ be a set.

I often use $\lvert X\rvert$ as the cardinarity of $X$.

I sometimes use $\# X$ when many vertical bars appear.

And I use $\mathscr{P}(X)$ as the power set of $X$.

I sometimes use $2^X$. It's useful to ask why we write $2^X$ for the power set.

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