Yes, they all work:
a = {1, 2, 3};
b = {1, 3, 2};
If[AllTrue[a, MemberQ[b, #] &] && AllTrue[b, MemberQ[a, #] &],
Print["a=b"], Print["a\[NotEqual]b"]]
If[Sort[a] === Sort[b], Print["a=b"], Print["a\[NotEqual]b"]]
a=b
a=b
But, can we always ensure that these methods are absolutely equivalent?