So the set of permutations for $\{x, y\}$ is $\{(x, y), (y, x)\}$. However, if I would try to make a set of permutations of an empty set $∅$, would the permutation set be $∅$ or $\{∅\}$?
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1If you're writing permutations, you should probably write ${(x, y), (y, x)}$ instead. As sets, you have ${x, y} = {y, x}$. – Aryaman Maithani Nov 01 '21 at 09:36
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@AryamanMaithani cleaned it up a bit, I'm not a mathematician by trade – Rick de Water Nov 01 '21 at 09:44
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There is exactly one permutation of $\emptyset$, which is $\emptyset$. Therefore, the set of all permutations is not $\emptyset$; it's $\{\emptyset\}$.
José Carlos Santos
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@Eric Because asserting that the answer is $\emptyset$ would mean that the set of all permutations of the empty set is empty. In other words, that the empty set has no permutations. But that's not true. It has one and only one permutation, which is $\emptyset$. – José Carlos Santos Mar 29 '23 at 08:32
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To me, your above comment feels just repeated your answer in a long way .. with no proof .. – Eric Mar 29 '23 at 09:26
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@JoséCarlosSantos isn't the correct answer {()} i.e. the set containing the 0-tuple (empty permutation)? – Lucubrator Oct 24 '23 at 12:23
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1I've seen it notated with epsilon as ε = () to distinguish it from the empty set, ∅ = {}. – Lucubrator Oct 24 '23 at 12:48