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Nullifying Dichotomies

Is it always possible to nullify a dichotomy? When tossing a coin, the outcome is either head or tail. However, head and tail are two faces of the same coin. In a sense, there isn't a split between head and tail as both belong to the same coin.

Let's try to think formally about this:

let A and B be two sets such that their elements follow distinct properties:

A={a | P(a)} , B={b | Q(p)}

A Dichotomy is the union of the two sets D = A ⋃ B

The dichotomy is nullified if a relation between P(a) and Q(b) exists.

The generation of a dichotomy can be synthesized as:

 D = {a, b | p R q}