Partition of a Set

Topics: Discrete Mathematics - Set

---

(definition)

Let $A$ be a set. We define the partition of a set $A$ (denoted $P$) such that it’s true that:

1. The elements of $P$ are non-empty subsets of $A$
2. The union of the elements of $P$ is $A$
3. The intersection of any pair of elements of $P$ is empty

Example 1

Let $2\mathbb{Z}=\{2n\mid n\in \mathbb{Z}\}$ and $\Pi =\{2n+1\mid n\in \mathbb{Z}\}$.

Both of these sets form a partition of $\mathbb{Z}$:

$$P=\{2\mathbb{Z},\Pi \}$$

Example 2

Let $D=\{0,1,\dots ,9\}$. This set has the partitions:

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

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

Those are just two instances of partitions of $D$. It can have other partitions.