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An ordered set is a set and a relation on that set satisfying certain properties. An ordered n-tuple is a list (colloquially) of n items in which order matters. If we take the set {a,b,c} with some order <=, then {b,a,c} with the same order is the same ordered set. If we take the ordered n-tuple (a,b,c), (b,a,c) is a different ordered n-tuple (except in the case where a = b).
As for uniqueness, the elements of a set have to be unique. Even if we ignore the ordering problem for a moment, you can't represent vectors as sets (in any nice way). The representation of (0,0,1) is {0,0,1}, which (by uniqueness) is equivalent to {0,1}. The representation of (0,1,1) is {0,1,1}, which is equivalent to {0,1}. Obviously that's problematic.