Partition of a set

Let $X$ be a set . A partition of $X$ is a set $\mathcal{P}$ of subsets of $X$ such that:

1. (Nonempty parts) For all $P \in \mathcal{P}$, $P \neq \emptyset$ (see empty set ).
2. (Disjointness) If $P,Q \in \mathcal{P}$ and $P\neq Q$, then $P \cap Q = \emptyset$ (using intersection ).
3. (Cover) $\bigcup_{P\in\mathcal{P}} P = X$ (using union ).

Partitions are equivalent data to equivalence relations : the parts are precisely the equivalence classes .

Examples:

• The residue classes modulo $3$ partition $\mathbb{Z}$ into $3\mathbb{Z}$, $1+3\mathbb{Z}$, and $2+3\mathbb{Z}$.
• The set $\mathbb{R}$ is partitioned by the two sets $(-\infty,0)$ and $[0,\infty)$.
• The singleton sets $\{\{x\} : x\in X\}$ form the discrete partition of $X$.