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Im struggling to understand whether the relation "is a permutation of" on N+

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I assume you mean 'sequences of natural numbers of length at least one' by 'N+'? What are your intuitions for those properties you mentioned? What have you tried and where are you struggling? – yatima2975 Dec 12 '12 at 20:58
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Hint: Go back to the definition. For symmetry, if you have a list of numbers $a_n$ such that $a_n\ R \ \mathbb N+$, is it also true that $\mathbb N+ \ R \ a_n$? You are given that for any given $k$, there is a unique $p$ such that $a_p=k$ from the definition of permutation. For anti-symmetry, can you show a permutation of $\mathbb N+$ that is not the identity? For reflexitivity, is your list a permutation of itself? For transitivity, given two permutations, can you compose them?

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I read this question as asking about the properties of the binary relation 'is a permutation of' with carrier set $\mathbb{N}^+$, not about the properties of the predicate 'is a permutation of $\mathbb{N}^+$' (both assuming the question asker is talking about sequences of natural numbers). The first part of your answer seems to use the second interpretation, while your hints for reflexivity and transitivity use the first. – yatima2975 Dec 12 '12 at 22:02
@yatima2975: I was thinking "is a permutation of $\mathbb N^+$" in all cases. For reflexivity and transitivity, I think all you have to say is that if two lists $a_n$ and $b_n$ have every number once and only once, they are related. – Ross Millikan Dec 12 '12 at 22:49

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