Equivalence Classes

Definition. [A set's partition] Consider a non-empty set $S$. A partition on $S$ is the collection (= set) of non-empty subsets $S_1, S_2, S_3, \dots, S_n$ of $S$ such that both of the following is true:

1. Every two sets $S_i$ and $S_j$ are disjoint: they share no common elements, and
2. The union of all the subsets is $S$ itself: $\bigcup S_i = S$.

Example: Let $S = \{0, 0.25, 0.5, 1, 1.5, -15, -15.67\}$. Each of the following is a partition on $S$:

• $P_1 = \{\{0, 1, -15\}, \{0.25, 0.5, 1.5, -15.67\}\}$.
• $P_2 = \{\{0\}, \{0.25\}, \{0.5\}, \{1\}, \{1.5\}, \{-15\}, \{-15.67\}\}$.
• $P_3 = \{\{0, 0.25, 0.5\}, \{1, 1.5\}, \{-15, -15.67\}\}$.

There are many other ways in which we can partition $S$.