Equivalence Classes & Quotients

Given an equivalence relation $R$ on $S$:
- The equivalence class of $x \in S$ is $[x]_R := \{y \in S \mid (x, y) \in R\}$.
- The quotient space is $S / R := \{[x]_R \mid x \in S\}$.
- The canonical projection $\pi_R: S \to S/R$ sends $x \mapsto [x]_R$.
- A system of representatives $M \subseteq S$ has exactly one element from
each equivalence class.

Key intuition: equivalence classes are bins. Each element of $S$ goes
into exactly one bin (the bin of all elements equivalent to it). $S/R$ is the
set of bins.

Worked example. Let $R$ on $\{1,2,3,4\}$ be: $\{1,2\}$ equivalent, $\{3,4\}$
equivalent. Then $S/R = \{\{1,2\}, \{3,4\}\}$, and a system of representatives
is $M = \{1, 3\}$.

WLOG (Latin sine detrimento', 'without loss') is a proof shorthand. We say 'WLOG assume $a \leq b$' when:
- The general statement is symmetric* in $a$ and $b$.
- We may swap their roles without affecting the conclusion.

It saves writing two cases. Examples:
- 'WLOG $x \leq y$' when the statement is symmetric in $x, y$.
- 'WLOG the set is non-empty' (the empty case is trivial).

WLOG is not valid when the asymmetry matters. Example: 'WLOG $a > 0$' is wrong if the statement talks about negative numbers specifically.