How to read $Q_j = \{k:\beta_k = j\}$?

Question

In my work, I came across the following expression:

$$Q_j = \{k:\beta_k = j\},\tag1$$

where $j = \mathcal{J} = \{1,...,J\}$ set of cell-phone tower,

$k = \mathcal{K} = \{1,...,K\}$ is a set of users, and

$\beta_k$ is the index of cell-phone tower to whom user $k$ is associated with.

I understand that equation $(1)$ expresses a set but, how to read it in terms of set theory?

Answer 1

1. $j = \mathcal{J} = \{1,...,J\}$ set of cell-phone tower
2. $k = \mathcal{K} = \{1,...,K\}$ is the set of users
3. $\beta_k$ is the index of cell-phone tower to whom user $k$ is associated with
4. $Q_j = \{k:\beta_k = j\}.\tag1$

Correction:

1. $\mathcal{J}=\{j_1,...,j_J\}$ is a set of cellphone towers.
2. $\mathcal{K}=\{k_1,...,k_K\}$ is a set of cellphone users.
3. For each cellphone user $k,\;\beta_k$ is the cellphone tower that $k$ is associated with.
4. $\forall j\in\mathcal J,\quad Q_j = \{k\in\mathcal{K}:\beta_k = j\}.\tag1$

$\mathcal J$ is a set of $J$ cellphone towers; $\mathcal K$ is a set of $K$ cellphone users; $$\forall j\in\mathcal J,\quad Q_j = \{k\in\mathcal{K}:k\text{ is associated with }j\}.$$ That last clause reads: for each tower $j$ in $\mathcal{J},\,Q_j$ is the subset of $\mathcal K$ such that every user in $Q_j$ is associated with $j.$