Set notation for 1) x_i and x_j belong to same class. 2) do not belong to the same class?

Question

Suppose have data X= {x₁, x₂,…,x_n } . Have a C_I ={ C₁,…,C_K,C_K+1 } , where { C₁,…,C_K,C_K+1 } represent K+1 classes/clusters

How write x_i and x_j belong to same class? Attempt C_I (x_i) = C_I (x_j)

How about x_i and x_j do not belong to the same class? Attempt C_I (x_i) ≠ C_I (x_j)

Answer 1

For $x_i$ and $x_j$ in the same class you could write $\exists C_l \in C_I$ such that $x_i,x_j \in C_l$ for some $$1\leq l \leq K+1.$$

Similarly, for $x_i$ and $x_j$ in a different class $\nexists C_l \in C_I$ such that $x_i,x_j \in C_l$ for some $$1\leq l \leq K+1.$$

If the classes in $C_I$ are not mutually disjoint, then $C_I(x_i) = C_I(x_j)$ is ambiguous as there may be some classes that contain $x_i$ and $x_j$ and some that contain just one of these elements. It is less ambiguous to say there exists a class where both of elements are contained in it.

Bear in mind notation is not universal and you can even use words instead if you’d like!