Definition 5.3. Let ${(A_{j})}_{j \in J}$ be a non-empty family of non-empty sets and $U$ an ultrafilter on $J$ .

Write $\prod_{j \in J} A_{j} ∕ U$ to be $\prod_{j \in J} A_{j} ∕ \sim$ (where $\sim$ is defined as in Definition 5.1).
${[{(a_{j})}_{j \in J}]}_{U}$ is the equivalence class of ${(a_{j})}_{j \in J}$ with respect to $\sim$ .
Let $B_{j} \subseteq A_{j}$ for every $j \in J$ . Then
${[{(B_{j})}_{j \in J}]}_{U} = {[{(a_{j})}_{j \in J}] \in \prod_{j \in J} A_{j} ∕ U : {j \in J : a_{j} \in B_{j}} \in U} .$