LƦ | Ʀamblings

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 \(P(q)\) and \(Q(b)\)

$$A = \{a | P(a)\}$$

$$B = \{b | Q(b)\}$$

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:

$$ D = {a, b | p R q} $$

2024-05-03: update

The dichotomy can be nullified only if \((A ⋃ B) ≠ U\)