Partition

Definition

A Family of Sets $P$ satisfies the following conditions, is a partition of $X$

• $\varnothing \notin P$
• $\bigcup _{A\in P}A=X$
• $\forall A,B\in P:A\ne B\Rightarrow A\cap B=\varnothing$

A grouping of its elements into non-empty subsets such that every element is included in exactly one subset.

Facts

${\sim }_P={\mathcal{R}}_P=X/P:=\{(x,y)|\exists A\in P,x,y\in A\}=\{(a,b)|a\in [b]\text{and}b\in [a],x,y\in A\}$ where $P$ is a partition, the result $\mathcal{R}$ is a Equivalence Relation, and $[\cdot ]$ is the element of the partition

The partition may be constructed from the Equivalence Relation