Partition Definition

$1$ point possible $($graded$)$

A partition of a set is a grouping of the set's elements into non$-$empty subsets, in such a way

that every element is included in one and only one of the subsets. In other words,

$C_1,C_2,$dots,$C_K$ is a partition of $\{1,2,$dots,$n\}$ if and only if

$C_1{C}_2$dots$C_K=\{1,2,$dots,$n\}$

and

$C_i{C}_j=\frac{O}{,}$ for any $ijin\{1,$dots,$k\}$

$($Union of all $C_j\text{'}$s is the original set and the intersection of any $C_i$ and $C_j$ is an empty set.$)$

For example,

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

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

$\{2,3,1\}$

are all partitions of the set $\{1,2,3\}.$