Let $n\in\mathbb{N}$, $E$ be a set, and consider the set $E^{[1,n]}$. I want to identify two elements $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ iff they are permutations of each other. What should be the equivalence relation in this case? I thought about $$(a_1,\ldots,a_n)\sim(b_1,\ldots,b_n):=(\exists\sigma)(\sigma\in S_n\ \land\ (\forall i)(1\leq i\leq n\implies a_i=b_{\sigma(i)})).$$ This passes reflexivity and transivity, but I am stuck on proving symmetry. Is this the right equivalence relation?
2026-04-07 23:18:25.1775603905
How to formalize notion of "up to ordering"
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