Partitions / Disjoint Unions (⊔)

A partition of a set $A$ is a set $$S = \{S_1, S_2, ..., S_n\}$$ such that

• $S_i \subseteq A$ and $S_i \neq \emptyset$ for $i= 1, 2, ..., n.$
• If $S_i \neq S_j$ then $S_i\cap S_j = \emptyset$ for all $i, j = 1,2, ..., n.$
• $S_1\cup S_2\cup S_3 \cup... \cup S_n=A.$

The set $A = \{a,b,c\}$ has all the following possible partitions: $$\{\{a\}, \{b\}, \{c\}\}$$ $$\{\{a\}, \{b,c\}\},$$ $$\{\{a, b\}, \{c\}\},$$ $$\{\{a, c\}, \{b\}\},$$ $$\{\{a, b, c\}\}.$$

The set of integer numbers may have the following partitions:

$$S = \{\{2k \mid k = 0, \pm 1, \pm 2, \pm 3, ...\}, \{2k + 1\mid k = 0, \pm 1, \pm 2, \pm 3, ...\}\};$$ $$S = \{\{3k \mid k = 0, \pm 1, \pm 2, \pm 3, ...\}, \{3k + 1\mid k = 0, \pm 1, \pm 2, \pm 3, ...\}, \{3k + 2 \mid k =0, \pm 1, \pm 2, \pm 3, ...\} \};$$ $$S = \{\{4k \mid k = 0, \pm 1, \pm 2, \pm 3, ...\}, \{4k + 1\mid k = 0, \pm 1, \pm 2, \pm 3, ...\}, \{4k + 2 \mid k =0, \pm 1, \pm 2, \pm 3, ...\},$$ $$\{4k + 3 \mid k =0, \pm 1, \pm 2, \pm 3, ...\}\};$$

and so forth.

Remark: If set $A$ has $n$ elements, how many possible partitions the set $A$ may have?