The definition of a hypercube is this:
The $n $-dimensional hypercube $Q_n$ is the graph with
$V = \left\{{ (e_1,\dots,e_n)|e_i \in \left\{{0,1}\right\}(i=1,\dots,n)}\right\}$in which two vertices are neighbours if and only if the corresponding rows differ in exactly one entry.
How can I write in set notation that I have a specific disjoint subset of the hypercube vertices, which is $\left\{{X,Y}\right\}$ $\left\{{C,V}\right\}$ where $X, Y, C,V$ are vertices $\left\{{e_1,e_2,\dots,e_i}\right\}$ where $e_i \in \{0,1\}$.
In other words, I want to say that out of all the hypercube $Q_n$ vertices $V$ I have isolated the disjoint subset $\left\{{X,Y}\right\}$ $\left\{{C,V}\right\}$ where each element of that subject is a vertex with $n$ binary digits
From a set theoretic standpoint, what you would want in terms of vertices is:
$$ \{ \{ a, b \} : a, b \subset V, |a| = |b| = 2, a \cap b = \emptyset\}$$