Number of Set Partitions by Number of Components/Examples/4 into 2

Example of Number of Set Partitions by Number of Components
A set with $4$ elements $\set {1, 2, 3, 4}$ can be partitioned into $2$ subsets in $\displaystyle {4 \brace 2} = 7$ ways:

Let $T$ be the set with $3$ elements $\set {1, 2, 3}$.

The partition of $T$ into $1$ set is:


 * $\set {1, 2, 3}$

to which $\set 4$ can be added in one way to achieve:


 * $\set {1, 2, 3 \mid 4}$

The partitions of $T$ into $2$ sets are:


 * $\set {1, 2 \mid 3}$
 * $\set {1, 3 \mid 2}$
 * $\set {2, 3 \mid 1}$

to which the element $4$ can be added in $2$ ways each:


 * $\set {1, 2, 4 \mid 3}$
 * $\set {1, 2 \mid 3, 4}$


 * $\set {1, 3, 4 \mid 2}$
 * $\set {1, 3 \mid 2, 4}$


 * $\set {2, 3, 4 \mid 1}$
 * $\set {2, 3 \mid 1, 4}$

Thus we have a total of $7$ ways of partitioning a $4$-element set into $2$ components.