📂Set Theory

Partition of a Set

Definition ¹

A partition of a set $X$ consists of all subsets $A,B,C$ of $X$ that satisfy the following conditions:

• (i): $$A,B \in \mathscr{P} \land A \ne B \implies A \cap B = \emptyset$$
• (ii): $$\bigcup_{C \in \mathscr{P} } C = X$$

Explanation

Although the mathematical expression might seem complex, simply put, it’s just about dividing the entire set into several parts without omission. If there’s leeway to delve into the mathematical definitions, it would be better to pay attention to details such as the partition $\mathscr{P}$ of $X$ being a subset of the power set $2^{X} = \mathscr{P} (X)$ of $X$.

As a simple example, consider the set of integers $\mathbb{Z}$. The set that includes the set of even numbers $2 \mathbb{Z} = \left\{ \cdots , -2 , 0 , 2 , \cdots \right\}$ and the set of odd numbers $1 + 2 \mathbb{Z} = \left\{ \cdots , -3 , -1 , 1 , 3 , \cdots \right\}$ becomes a partition of $\mathbb{Z}$. Here, $\mathscr{P} \subset 2^{\mathbb{Z}}$ is a set that contains subsets of $\mathbb{Z}$, and the number of elements is $2$. It’s good to practice understanding exactly what belongs where and whether it is an element or a set, according to the definition, without glossing over.