disjoint Partition of sets

Question

Hello I'm a bit confused by what this definition means

$$DT_n = \{ \{C,D \} \mid C,D ⊆ S_n \text{ and } C \cup D = S_n \text{ and } C \cap D = \emptyset \} $$ where $S_n = \{1,2....n\}$ and n is a natural number. I'm given the examples of $DT_0 = \{\{\emptyset, \emptyset \}\}$and $DT_1=\{\{\{1 \}, \emptyset \}\}$. But I'm still a bit confused on how one would go about continuing the sets for say n = 2, or n = 3. Would $DT_2 = \{\{\{1\}, \emptyset\}, \{\{2\}, \emptyset\}\}$ and how would DT_3 work? Would that be sets of sets of two elements? I don't really understand the notation.

Any help would be appreciated! Thanks!

Answer 1

$DT_n$ is a set. Each element of the set is a set containing two sets. Those two sets partition $S_n$. $DT_n$ consists of all the partitions of $S_n$ into two disjoint sets.

Answer 2

DT_3 has 4 two set partitions, one with empty set in it and three with a singleton in it.