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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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    See http://math.stackexchange.com/a/129303/8562012-04-11
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    @Rahul Interesting.2012-04-11

2 Answers 2

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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 $A\to B$.

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    Why is it called $2$?2012-04-11
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    @Peter, see http://en.wikipedia.org/wiki/Natural_number#Constructions_based_on_set_theory2012-04-11
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    @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.2012-04-11
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    @jwodder Thanks!2012-04-11
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Motivation: the cardinality is $2^{|A|}$.