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