Re: ordered pairs/n-tuples as collections of sets

The best I can see is to do this:
Define S=the set of all 1-element sets.
i.e. S={{x}:x is an element of X}, where X is the system containing the
values of your coordinates (e.g. if you are looking at Euclidean
cooredinates for the plane, X=R)

Next, define T=(a,b)={{a},{{b}}}. Note that this removes the problem of
a=b.

Then T_1=T intersect S, and T_2=T-S

If you need to keep the same definition of T, then you can still use
this definition, but not that you need to interpret the empty set as
meaning "the same as T_1", so if, for example:
T=(a,a), then by definition T={{a}}
so T_1={a} and T_2={} (using my definitions)

.

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