# Set of subsets notation. [duplicate]

Why is it that we denote the set of all subsets of $A$ by $2^A$?

Is there any historical or logical cause that motivated this notation?

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## marked as duplicate by Pedro Tamaroff♦, amWhy, Andrey Rekalo, Ayman Hourieh, L.G.Jul 9 '13 at 19:10

–  Rahul Apr 11 '12 at 2:04
@Rahul Interesting. –  Pedro Tamaroff Apr 11 '12 at 2:06

Another reason is that the set of all subsets of $A$ can be identified with all functions $A\to \{0,1\}$ and $\{0,1\}$ is sometimes called $2$. Plus the common usage of $B^A$ to denote the set of all functions $A\to B$.
Why is it called $2$? –  Pedro Tamaroff Apr 11 '12 at 2:11
@PeterT.off: In pure set theory, each natural number is defined as the set of all smaller natural numbers, and so $0=\emptyset$, $1 = \{0\} = \{\emptyset\}$, $2 = \{0,1\} = \{\emptyset, \{\emptyset\}\}$, etc. –  jwodder Apr 11 '12 at 2:18
Motivation: the cardinality is $2^{|A|}$.