Given a set $$ A_{i} \subseteq S | \forall i \in \mathbb{N}, \exists j,k < i
A_k, A_j \subseteq {A}_{i}= S | A_k \cup A_j = S \text{ and }A_k \cap A_j = \varnothing
Given a set $$ A_{i} \subseteq S | \forall i \in \mathbb{N}, \exists j,k < i
A_k, A_j \subseteq {A}_{i}= S | A_k \cup A_j = S \text{ and }A_k \cap A_j = \varnothing