Set of subsets notation.

Question

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?

See http://math.stackexchange.com/a/129303/856 – Apr 11 '12 at 02:04 — , Apr 11 '12 at 02:04

score 6 · Answer 1 · answered Apr 11 '12 at 01:59

6

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

answered Apr 11 '12 at 01:59

anon

151,657

score 6 · Accepted Answer · answered Apr 11 '12 at 02:01

6

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 $A\to B$.

answered Apr 11 '12 at 02:01

lhf

216,483

Why is it called $2$? – Pedro Apr 11 '12 at 02:11
@Peter, see http://en.wikipedia.org/wiki/Natural_number#Constructions_based_on_set_theory – lhf Apr 11 '12 at 02:17
3

@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 02:18
@jwodder Thanks! – Pedro Apr 11 '12 at 02:20

Set of subsets notation.

2 Answers2