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This question is related to Notation for all permutations of a set.

I am looking for notation for a set comprising the union of all possible permutations of each element of a power set.

For example, if $A = \{a,b,c\}$, then $\mathcal{P}(A) = \{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$.

But I am looking for:

$\{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{b,a\}\{a,c\},\{c,a\},\{b,c\},\{c,b\},\{a,b,c\},\{a,c,b\},\{b,a,c\},\{b,c,a\},\{c,a,b\},\{c,b,a\}\}$

Henno Brandsma
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guero64
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    The order of the elements in a set is irrelevant ${a,b}={b,a}$ so your second set has many duplicates. And duplicates don’t matter either. Both sets are the same. – Henno Brandsma Jan 31 '20 at 18:36
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    Now... if you were to instead talk about the elements in the set you are describing as sequences instead of sets... so ${\emptyset,a,b,c,ab,ac,ba,bc,ca,cb,abc,acb,\dots}$ you might be able to get away with something like $\bigcup\limits_{n=0}^\infty A\frac{[n]}{~}$ where here we use the notation that $Y\frac{X}{~}$ is the set of injective functions from $X$ to $Y$ and the notation that $[n]$ is the prototypical $n$-element set (be that ${1,2,3,\dots,n}$ or ${0,1,2,\dots,n-1}$, it doesn't matter which for our purposes). – JMoravitz Jan 31 '20 at 19:50

1 Answers1

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For your example:
{empty set} $\cup$ { {a} : a in A } $\cup$
{ (a,b) : a,b in A, a $\neq$ b) } $\cup$
{ (a,b,c) : a,b,c in A, a $\neq$ b, b $\neq$ c, c $\neq$ a }.

Ordered pairs and ordered triplets are needed because,
for example, {a,b} = {b,a} while (a,b) $\neq$ (b,a).

William Elliot
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