Partition of a Set

Partition of a set

The partition of a set $A$ is a collection of subsets of $A$ such that none of the subsets are empty that is no two subsets in the collection have common elements, and the union of all the subsets in the collection is equal to $A$. If we denote a partition of a set $A$ by $\cal{P}$, then by definition, we can derive the following facts:

• $B \in {\cal P} \Rightarrow B \ne \emptyset$

• $(B\in {\cal P} \land C \in {\cal P}) \Rightarrow B\cap C = \emptyset$

• $\bigcup \limits_{B\in {\cal P}}B = A$

Here, the notation $\bigcup \limits_{B\in {\cal P}}B$ refers to the union of all the sets in the partition ${\cal P}$. Let’s look at a few examples to further understand how to partition a set.

Examples

Let’s take a set $D =\{0,1,2,3,4,5,6,7,8,9\}$ as a first example. We’ll carefully verify that each of the following is a partition of the set $D$:

$\cal P_1 = \{\{0,2,4,6,8\},\{1,3,5,7,9\} \}$

$\cal P_2 = \{\{0\}, \{1,2,3,4,5,\},\{6,7,8,9\} \}$

$\cal P_3 = \{\{0,1,3,6,9\},\{2,4,5,7,8\} \}$ ...