Equivalence Classes

Examples:

• $P_1 = \;$$\{[0], [0.25]\} \;$$= \{\{0, 1, -15\}, $$\;\{0.25, 0.5, 1.5, -15.67\}\}$ would correspond to an equivalence relation that relates integers to integers, and fractions to fractions. We can call it $\text{Int-vs-Frac}$.
• $P_2 =$$\; \{\{0\}, $$\;\{0.25\}, $$\;\{0.5\}, $$\;\{1\}, $$\;\{1.5\}, $$\;\{-15\}, $$\;\{-15.67\}\}$ corresponds to the $\text{Equals}$ relation on $S$.
• $P_3 \;$$= \{[0], [1], [-15]\} \;$$= \{\{0, 0.25, 0.5\}, $$\;\{1, 1.5\}, $$\;\{-15, -15.67\}\}$ corresponds to the $\text{Same-Int-Parts}$ relation on $S$.

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Fun Class Activity: Let $S = \{1, 1.23, 2, $$\;2.0, 2.5, 3, \frac{9}{3}\}$.

1. Write out the partition on $S$ that corresponds to the relation $\text{Same-$\mathbb{Q}$-Num}$.
2. What relation does the following partition of $S$ correspond to: $\{\{1, 1.23\}, $$\;\{2, 2.0, 2.5\}, $$\;\{3, \frac{9}{3}\}\}?$