Equivalence Classes

Definition. [Equivalence Classes] Let $R$ be an equivalence relation on a set $S$, and consider some object $x \in S$. The set $$[x] = \{y \in S \mid (x, y) \in R\}$$ (

$[x] = \{y \in S \mid (x, y) \in R\}$

) (that is, the set of all the elements $y$ of $S$ that relate to $x$ through $R$) is called the equivalence class, or equivalence set, containing $x$.

Example: In the relation $\text{Same-$\mathbb{Q}$-Num}$ from slide 19, since we have $\frac{1}{2} \;$$= \frac{2}{4} \;$$= 0.5 \;$$= 50:100 \;$$= 50\%$, we could write:

$[\frac{1}{2}] \;$$= \{\frac{1}{2}, \,$$\frac{2}{4}, \,$$0.5, \,$$50:100, \,$$50\%, \,$$\dots\}$

You can also write $[0.5]$, $[50\%]$, $[50:100]$, etc., instead of $[\frac{1}{2}]$.