Equivalence Classes

The following sets, however, aren't partitions of $S$:

• $NP_1 =$$\; \{\{0, 1, -15\}, $$\;\{0.25, 0.5, 1, $$\;1.5, -15.67\}\}$ because $1$ appears in more than one subset.
• $NP_2 =$$\; \{\{0\}, $$\;\{0.25\}, $$\;\{1\}, $$\;\{1.5\}, $$\;\{-15\}, $$\;\{-15.67\}\}$ because $0.5$ is missing.
• $NP_3 =$$\; \{\{0, 0.25, 0.5\}, $$\;\{\}, $$\;\{1, 1.5\}, $$\;\{-15, -15.67\}\}$ because the 2nd item is an empty set $\{\}$, but partitions mustn't contain empty sets.

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Theorem. [Equivalence classes vs. partitions] Let $R$ be an equivalence relation on a set $S$. The following are true:

1. The set of all the equivalence classes under $R$ is a partition on $S$.
2. Every partition on $S$ corresponds to some equivalence relation on $S$.