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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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marked as duplicate by Pedro Tamaroff, amWhy, Andrey Rekalo, Ayman Hourieh, Start wearing purple Jul 9 '13 at 19:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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

2 Answers 2

up vote 6 down vote accepted

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 $2$? – Pedro Tamaroff Apr 11 '12 at 2:11
@Peter, see… – lhf Apr 11 '12 at 2:17
3 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
@jwodder Thanks! – Pedro Tamaroff Apr 11 '12 at 2:20

Motivation: the cardinality is $2^{|A|}$.

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