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 $S$.